Geometry Practice: Examples and Practice (Middle/Upper Grades) 1580373275, 9781580377522

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About the Authors Barbara Sandall—Ed.D., Illinois State University; MA, DePaul University; BA, Illinois Wesleyan University—is currently assistant professor in Curriculum and Instruction at Western Illinois University teaching Elementary Science Methods, Physical Science for Elementary Teachers, and other courses for elementary education majors. Dr. Sandall taught middle-school science for 18 years, a fifth-grade selfcontained class for one year, as well as courses in elementary music and basic computer programming. She has also worked at Illinois State University in the Center for Mathematics, Science, and Technology Education; the Eisenhower Midwest Consortium for Mathematics and Science Education at the North Central Regional Educational Laboratory (NCREL); and the Teachers Academy for Mathematics and Science in Chicago. Dr. Sandall is the past treasurer for the Illinois Science Teachers association and is currently a Regional Director for the National Science Teachers Association. Melfried Olson—Ed.D., Oklahoma State University; MA, University of Arkansas; BS, Valley City State College—is currently a mathematical researcher at the Curriculum Research & Development Group at the University of Hawaii. Dr. Olson taught mathematics content and methods courses for 18 years at Western Illinois University and 11 years at the University of Wyoming. He taught grades 7–12 for three years, teaching grade 7, grade 8, Algebra I, Geometry, Algebra II, and Pre-Calculus. Dr. Olson has been active at the local, state, and national level in mathematics education through numerous publications, presentations, and grants. Dr. Olson has been an active consultant in mathematics teaching and learning. He directed a nationwide staff development project in geometry as well as extensive staff development efforts with teachers K–12. For three years, Dr. Olson served as a co-editor for the Problem Solvers section of the NCTM journal Teaching Children Mathematics. He served as president for the Research Council on Mathematics Learning and on the board of directors for the Illinois Council of Teachers of Mathematics and School Science and Mathematics. Travis Olson has a master’s degree in mathematics from Western Illinois University, has taught mathematics (intermediate algebra through calculus III) two years at Cottey College, and is currently a doctoral student in mathematics education at the University of Missouri—Columbia. He has assisted with numerous staff development activities for elementary and middle-school teachers of mathematics and has taught two years in the Johns Hopkins University Center for Talented Youth program.

Mathematical consultant: Joseph A. Kunicki received his BS (1986) and MS (1988) in mathematics from Youngstown State University in northeast Ohio as a non-traditional student. His pre-academic career included five years in the United States Marine Corps, working as an aviation electrician, and 15 years as a blue-collar worker, predominantly in the Bay Area of northern California. Dr. Kunicki finished his Ph.D. in mathematics education at the Ohio State University in 1994, where he concentrated on multi-variate statistical analyses, abstract algebra, and college-level mathematics pedagogy. He has substantive experience working with the National Council of Mathematics Teachers directed toward the Curriculum & Evaluation Standards and their alignment with various district and state mathematics standards. Dr. Kunicki is currently an associate professor of Mathematics at the University of Findlay in northwest Ohio. He resides in Arcadia, Ohio, with his wife Pat St. Onge, a graphic illustrator for Mark Twain Media, Inc., and their two Siberian Huskies, Star and Sherman.

Geometry Practice By Dr. Barbara Sandall, Ed.D. Dr. Melfried Olson, Ed.D. Travis Olson, MS

COPYRIGHT © 2006 Mark Twain Media, Inc. ISBN 978-1-58037-752-2 Printing No. 404044-EB Mark Twain Media, Inc., Publishers Distributed by Carson-Dellosa Publishing Company, Inc. The purchase of this book entitles the buyer to reproduce the student pages for classroom use only. Other permissions may be obtained by writing Mark Twain Media, Inc., Publishers. This product has been correlated to state, national, and Canadian provincial standards. Visit www.carsondellosa.com to search and view its correlations to your standards. All rights reserved. Printed in the United States of America.

Table of Contents

Geometry Practice

Table of Contents Introduction to the Math Practice Series.....................................................1

Chapter 5: Triangles..................................51 Parts of a Triangle, Similar and Congruent Triangles, Properties of Triangles, Perimeter and Area.........51 Practice.................................................57 Challenge Problems..............................60 Checking Progress................................62

Common Mathematics/Geometry Symbols and Terms..............................................2 Chapter 1: Introduction To Geometry.....13 Shapes, Congruence, Similarity, and Symmetry........................................13 Practice.................................................16 Challenge Problems..............................18 Checking Progress................................19

Chapter 6: Polygons and Quadrilaterals......................................63 Polygons and Quadrilaterals.................63 Practice.................................................71 Challenge Problems..............................73 Checking Progress................................75

Chapter 2: Coordinate and Non-coordinate Geometry.............................................21 Midpoint Formulas, Slope, Equations, Distance...........................................21 Practice.................................................27 Challenge Problems..............................31 Checking Progress................................32

Chapter 7: Circles.....................................76 Radius, Diameter, Circumference and Area, Chords, Tangents, Secants, Arcs, Inscribed Angles, Finding the Equation of a Circle.........................76 Practice.................................................81 Challenge Problems..............................87

Chapter 3: Angles.....................................33 Measuring and Classifying Angles........33 Practice.................................................38 Challenge Problems..............................40 Checking Progress................................43

Geometry Check-Up.................................88 Practice Answer Keys..............................93

Chapter 4: Patterns and Reasoning........44 Patterns, Structure of Geometry, Conditional Statements, and Deductive Reasoning.......................44 Practice.................................................46 Challenge Problems..............................48

Check-Up Answer Keys.........................118 References..............................................122

Checking Progress................................50

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Geometry Practice

Introduction to the Math Practice Series

Introduction to the Math Practice Series The Math Practice series of books will introduce students in middle school and high school to the course topics of Pre-algebra, Algebra, Algebra II, and Geometry. The content of all of the practice books are aligned with the National Council of Teachers of Mathematics (NCTM) Principles and Standards for School Mathematics. (NCTM 2000) This series is written for classroom teachers, parents, families, and students. The practice books in this series can be used as a full unit of study or as individual lessons to supplement textbooks or curriculum programs. Parents and students can use this series as an enhancement to what is being done in the classroom or as a tutorial at home. Students will be given a basic overview of the concepts, examples, practice problems, and challenge problems using the concepts introduced in the section. At the end of each section, there will be a set of problems to check progress on the concepts and a challenge set of problems over the whole section. At the end of the book, there will be problems for each section that can be used for assessment. According to the Mathematics Education Trust and NCTM, new technologies require the fundamentals of algebra and algebraic thinking be a part of the background for all citizens. These technologies also provide opportunities to generate numerical examples, graph data, analyze patterns, and make generalizations. An understanding of algebra is also important because business and industry require higher levels of thinking and problem solving. NCTM also suggests that understanding geometry, including the characteristics and properties of two- and three-dimensional shapes, spatial relationships, symmetry, the use of visualization, and spatial reasoning, can also be used in solving problems. The NCTM Standards suggest content and vocabulary are necessary, but of equal importance are the processes of mathematics. The process skills described in the Standards include: problem solving, reasoning, communication, and connections. The practice books in this series will address both the content and processes of algebra and algebraic thinking and geometry. This practice book, Geometry Practice, will help students practice geometric concepts.

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Geometry Practice

Common Mathematics/Geometry Symbols and Terms

Common Mathematics/Geometry Symbols and Terms Term

Symbol/Definition

Undefined Terms

Cannot be defined but can be described

Point

Specific place in space. It has A no dimension, length, width, or depth. It is represented by a dot and Point A labeled by a single capital letter.

Line

An infinite set of points that extends A B C continuously and without limit in Line AB, AC, or BC either direction. ↔ ↔ ↔ Also written AB, AC, BC

Plane

A flat surface containing an infinite number of points with no boundaries and no thickness; when a plane is represented by a four-sided figure, a capital letter is placed in one corner as a label.

Example Point

A

Line Segment

A line segment is part of a line with two endpoints. Two letters with a A B bar over them represent the line Segment AB, or written AB segment.

Ray

A ray has one endpoint and extends infinitely in the opposite direction from A B C the endpoint. Rays are identified with arrows on top of the letters showing Ray AC or BC both the endpoint and the direction → → Also written AC, BC in which the rays are going.

Opposite Rays

Two rays traveling in opposite A B C directions that have the same endpoint and form a straight line Rays BA and BC are opposite rays.

→

→

Also written BA, BC

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Geometry Practice

Common Mathematics/Geometry Symbols and Terms

Common Mathematics/Geometry Symbols and Terms (cont.) Term

Symbol/Definition

Parallel Lines

Two lines in the same plane that do not intersect Symbol: ll

Example A

B

C

D AB ll BC

Collinear

Points that lie on the same line A

B

C

D A, B, and C are collinear. Non-collinear Points

Points that do not lie on the same line

A

B

D

E A, B, D, and E are noncollinear because they cannot all lie on the same line simultaneously. Angle

A figure formed by two rays that have a common endpoint; the rays are the sides of the angle

Vertex of an Angle

The common endpoint where two rays meet; the plural form of vertex is vertices.

Polygon

Closed figure in a plane, formed by three or more segments called sides. The endpoints where the segments meet are called the vertices of the polygon.

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Vertex

Common Mathematics/Geometry Symbols and Terms

Geometry Practice

Common Mathematics/Geometry Symbols and Terms (cont.) Term

Symbol/Definition

Congruent Shapes

Two figures are congruent if they match exactly in both size and shape.

Perimeter of a Polygon

The sum of the measures of all of the sides, expressed in linear units such as inches, feet, meters, centimeters, miles, etc.

Area

Example

The distance around the polygon.

If each side is 1 cm, then the perimeter is 4 cm.

The number of square units that the plane figure contains, expressed in square inches, square centimeters, etc. A measure of the interior region contained within the polygon.

If each side is 1 cm, then the area is side x side, so the area is 1 x 1 = 1 cm 2.

Ratio

The quotient when one number is divided by another number; mostly written in fraction form

If the numbers are represented by a and b, then the ratio of the two numbers is a or a divided by b as long , b as b ≠ 0.

Proportion

An equation that sets two ratios equal to each other

a c = b d

Postulate

A mathematical statement that is accepted as true without any proof

Two points determine a single line. Three non-collinear points not on the same line are needed to determine a single plane. If two planes intersect, they intersect in exactly one line.

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Common Mathematics/Geometry Symbols and Terms

Geometry Practice

Common Mathematics/Geometry Symbols and Terms (cont.) Term

Symbol/Definition

Example

Theorem

A mathematical statement that can logically be proven true using deductive reasoning.

If two lines intersect, then they intersect at exactly one point. Through a line and a point not on that line, there is only one plane.

Geometric Proofs

A proof is a series of reasons, that lead logically from the given conditions to the desired conclusion. The reasons can be definitions, postulates, or previously proven theorems. The first step in the writing of proofs is to convince yourself that the statement is true. The second is to write them in a logical way that would convince a reader.

Inductive reasoning starts with examples, and then the examples are used to generate conjectures. Deductive reasoning is reasoning that uses definitions, theorems, and postulates to prove a new theorem true. In deductive proofs, compare the statements with the reasons. In deductive proofs, ask yourself: What do you know? What can you infer? What can you conclude?

Addition Sign Subtraction Sign

+

2+2=4

­­–

4–2=2

Multiplication Sign

x or a dot • or a number and a letter(s) together or parentheses

3x2 2•2 2x 2(2)

Division Sign

÷ or a slash mark ( / )

6÷2



or a horizontal fraction bar

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4/2

$s

Common Mathematics/Geometry Symbols and Terms

Geometry Practice

Common Mathematics/Geometry Symbols and Terms (cont.) Term

Symbol/Definition

Example

Equals or is equal to

= ≠ < > > < @

2 + 2 = 4



5 ≠ 1



2 < 4



4 > 2



2 + 3 ≥ 4, 2 • 5 ≥ 10



Does Not Equal Less than Greater than Greater than or equal to Less than or equal to Congruent Perpendicular

2 + 1 ≤ 4, 3 + 2 ≤ 5 DABC @ DDEF A

^

C

B

D

Line segment AB is perpendicular to line segment CD or ­ —­ — written as AB ^ CD. Pi

A number that is approximately SuS or 3.14, represented as π

3.1415926...

Similar

~

DABC ~ DDEF

Arc

Part of a circle

B

Arc AB or written as (

A

Square

Quadrilateral with four congruent sides and four right angles

Rectangle

Quadrilateral with four right angles

Parallelogram

Quadrilateral whose opposite sides are parallel

Quadrilateral

Polygon with four sides

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AB

Geometry Practice

Common Mathematics/Geometry Symbols and Terms

Common Mathematics/Geometry Symbols and Terms (cont.) Term

Symbol/Definition

Pentagon

Polygon with five sides

Hexagon

Polygon with six sides

Octagon

Polygon with eight sides

Coordinates/ Coordinate Plane

A coordinate plane is a twodimensional grid used in graphing. Points in the plane can be matched one-to-one with ordered pairs of real numbers, called the coordinates. The horizontal line is called the xaxis, and the vertical line that is perpendicular to the x-axis is called the y-axis.

Example

y-axis 4 3 2 1 -4 -3 -2 -1 0 1 2 3 4

x-axis

-1 -2 -3 -4

Ordered Pairs

Describes a point in the coordinate plane. The first number of the pair, the x-coordinate, tells the location relative to the x-axis. The second number, the y-coordinate, tells the location relative to the y-axis.

(3, 8) means 3 to the right of 0 on the x-axis and 8 up on the y-axis. The point is where these two numbers intersect.

Midpoint

The midpoint of a line segment is the single point that is an equal distance from both endpoints.

The midpoint of a line segment is x1 + x2 , y1 + y2 2 2

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(

)

Common Mathematics/Geometry Symbols and Terms

Geometry Practice

Common Mathematics/Geometry Symbols and Terms (cont.) Term

Symbol/Definition

Example

Linear Equation

ax + by = c, where a, b, and c are constants, and x and y are variables.

2x – 3y = 6 -0.4x + 1.2y = -1

Angle Bisector

A ray that bisects the angle

Obtuse Angle

An angle that is greater than 90°

Acute Angle

An angle that is less than 90°

Right Angle

An angle that is equal to 90°

Vertical Angle

Four angles are formed when two lines intersect. The opposite angles always have the same measure, and they are called vertical angles.

Supplementary Angles

If two angles form a straight line, the sum of the two angles is 180°.

Complementary Angles

When the sum of two angles is 90°

Adjacent Angles

Angles that share a common side

∠ABD is adjacent to ∠DBC D A

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B

C

Common Mathematics/Geometry Symbols and Terms

Geometry Practice

Common Mathematics/Geometry Symbols and Terms (cont.) Term Reflex Angle

Symbol/Definition

Example

A reflex angle is greater than 180° but less than 360°. The arc at the vertex indicates that it is a reflex angle.

Straight Angle

Forms a straight line

Protractor

Tool used to measure angles

Symmetry

If a figure can be divided into two parts, each of which is a mirror image of the other, then it has line symmetry, or reflection symmetry.

Equation

Mathematical sentence in which two phrases are connected with an equals (=) sign

5 + 7 = 12 3x = 12 1=1

Absolute Value

The absolute value of a number can be considered as the distance between the number and zero on a number line. The absolute value of every number will be either positive or zero. Real numbers come in paired opposites, a and -a, that are the same distance from the origin but in opposite directions.

Absolute value of a : Ia I = a if a is positive. Ia I = a if a is negative. Ia I = 0 if a = 0. With 0 as the origin on the number line, the absolute value of both -3 and +3 is equal to 3, because both numbers are three units in distance from the origin.

-3 -2 -1 0 1 2 3 Circumference

The distance around a circle

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C = πd C = 2πr d = diameter; r = radius

Geometry Practice

Common Mathematics/Geometry Symbols and Terms

Common Mathematics/Geometry Symbols and Terms (cont.) Term Diameter

Radius

Symbol/Definition The measurement of a straight line passing through the center of a circle, or 2 x radius (2r)

The diameter (d) of this circle is 4 cm.

4 cm

One-half the diameter of a circle, or

s! x diameter (s! d )

Area of a Circle

Example

The radius (r) of this circle is 2 cm.

2 cm

The measure of the interior of a

A = πr 2 A=π d 2

( )

circle

2

d = diameter; r = radius Distance

The length of a line segment; given two points in a plane, (x1, y1) and (x2, y2), the distance between them is given by the formula:

Slope

Given the points: (-2, 3) and (3, 1), the distance is (-2 √ – 3)2 + (3 – 1)2 =

(x √ 1 – x 2)2 + (y1 – y2)2

25 √ + 4 = 29 √

The measure of the incline of a

To find slope: Slope-intercept 3x + y – 2 = 0 Solve for y : y = -3x + 2 The number in front of the x (-3) is the slope; the 2 is the y-intercept.

line

Two-Point Method: y – y 1 Slope = 2 x 2 – x 1 Given two points on the line: (3, 2) and (5, 1) 1 – 2 -1 Slope = = 5 – 3 2 © Mark Twain Media, Inc., Publishers

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Geometry Practice

Common Mathematics/Geometry Symbols and Terms

Common Mathematics/Geometry Symbols and Terms (cont.) Term

Symbol/Definition

Example

Triangle

A three-sided polygon

D ABC

A C

B Isosceles Triangle

A triangle that has two congruent angles and two congruent sides

Equilateral Triangle

A triangle that has three equal sides (and angles)

Acute Triangle

A triangle in which every angle is less than 90°

Obtuse Triangle

A triangle in which one angle is greater than 90°

Right Triangle

A triangle in which one angle is equal to 90° 90°

Equiangular Triangle

A triangle in which all angles

60°

(and sides) are equal; same as 60°

equilateral triangle Scalene Triangle

60°

A triangle in which there are no congruent sides or angles

Hypotenuse

The side opposite the right angle in a right triangle

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Hypotenuse

Common Mathematics/Geometry Symbols and Terms

Geometry Practice

Common Mathematics/Geometry Symbols and Terms (cont.) Term

Symbol/Definition

Legs

The other two sides of a right triangle, adjacent to the hypotenuse and forming the right angle

Pythagorean Theorem

Example

The square of the hypotenuse of a right triangle is equal to the sum of the squares of the legs

Legs

Let h represent the hypotenuse, s1 represent one leg or side and s2 represent the other leg or side. h 2 = s 12 + s 2 2 h 2 = 32 + 22 h 2 = 13 h = 13

Similar Triangles

Congruent Triangles

h

h

3 in. (s1)

2 in. (s2)

Triangles that have congruent angles and corresponding sides that are proportional in length. Triangles that have corresponding sides and angles equal.

Median

A segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side

Median

Altitude

A segment from a vertex that is perpendicular to the opposite side or to the line that is an extension of the opposite side

Altitude

Perpendicular Bisector

A line segment, a ray, or a plane that is perpendicular to a given segment and bisects it

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Chapter 1: Introduction to Geometry

Geometry Practice

Name:

Date:

Chapter 1: Introduction to Geometry Basic Overview: Shapes, Congruence, Similarity, and Symmetry Polygons have names determined by the number of sides: triangles have three sides, quadrilaterals have four sides, pentagons have five sides, hexagons have six sides, and octagons have eight sides. The polygons above are in one plane or surface. Objects can also be solid and three-dimensional and be in more than one plane. Two figures are congruent if they have the same shape and the same size. If two figures are similar, their sides are proportional and corresponding angles are congruent. Geometric figures are similar if figures have the same shape. They may or may not have the same size. If a figure can be divided into two parts, each of which is a mirror image of the other, it has line symmetry, or reflection symmetry. A figure in a plane has a line of symmetry if it can be mapped onto itself by a reflection. The lines of symmetry can, but need not be, vertical or horizontal. Figures can also have rotational symmetry if the figure can be mapped on itself by rotating the figure. Translational symmetry is created when an object is moved without rotating or reflecting it. Glide reflection symmetry is a transformation that consists of a translation followed by a reflection with the line of translation also essentially serving as the line of symmetry. Examples of Shapes, Congruence, Similarity, and Symmetry Examples of Polygons in One Plane:

Examples of Three-dimensional Shapes:





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Chapter 1: Introduction to Geometry

Geometry Practice

Name:

Date:

Chapter 1: Introduction to Geometry (cont.) Examples of Congruence:

Examples of Similarity:

Examples of Symmetry:

Line Symmetry:

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Each dashed line can be a line of symmetry.

14

Chapter 1: Introduction to Geometry

Geometry Practice

Name:

Date:

Chapter 1: Introduction to Geometry (cont.) Rotational Symmetry:



Rotating 180° TOP

Rotating 120°

BOTTOM



TOP

Translational Symmetry:

B





B

Glide Reflection Symmetry:

R R © Mark Twain Media, Inc., Publishers

R 15

B

A



C

B

BOTTOM

C

A

Chapter 1: Introduction to Geometry

Geometry Practice

Name:

Date:

Chapter 1: Introduction to Geometry (cont.) Practice: Shapes, Congruence, Similarity, and Symmetry Directions: Draw an example of the polygons below. Explain why your drawing shows an example of the polygon. 1.













Octagon Drawing:















2.

Hexagon









Drawing:































3.

Triangle









Drawing:































4.

Quadrilateral







Drawing:



































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Explanation:



Explanation:

Explanation:

Explanation:

16

Chapter 1: Introduction to Geometry

Geometry Practice

Name:

Date:

Chapter 1: Introduction to Geometry (cont.) 5.

Which figures are similar and which are congruent?



Similar:





A.







C.







Congruent:







B.







D.







Directions: Divide the shaded regions into two congruent parts. Show a different possible solution in each of the regions below. 6.













7.

8.













9.

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Chapter 1: Introduction to Geometry

Geometry Practice

Name:

Date:

Chapter 1: Introduction to Geometry (cont.) Challenge Problems: Shapes, Congruence, Similarity, and Symmetry 1.

Draw a figure that has line symmetry and is different from the examples given.

2.

Draw a figure that has rotational symmetry and is different from the examples given.

3.

Draw a figure that shows translational symmetry and is different from the example given.

4.

Draw a figure that shows glide reflection symmetry and is different from the examples given.

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Chapter 1: Introduction to Geometry

Geometry Practice

Name:

Date:

Chapter 1: Introduction to Geometry (cont.) Checking Progress: Shapes, Congruence, Similarity, and Symmetry 1.

Draw a figure of a polygon in one plane.

2.

Draw a figure of a polygon in more than one plane.

3.

Draw a figure that shows 180° symmetry.

4.

Draw a figure with horizontal symmetry.

5.

Draw a figure that has vertical symmetry.

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Chapter 1: Introduction to Geometry

Geometry Practice

Name:

Date:

Chapter 1: Introduction to Geometry (cont.) Complete the table below.

Polygon

6.

Hexagon

7.

Square

8.

Circle

9.

Quadrilateral

10.

Octagon

11.

Pentagon

12.

Triangle

13.

Rectangle

Drawing



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Number of Sides

Chapter 2: Coordinate and Non-Coordinate Geometry

Geometry Practice

Name:

Date:

Chapter 2: Coordinate and Non-coordinate Geometry Basic Overview: Midpoint Formulas, Slope, Equations, Distance The midpoint of a line segment is the single point that is equal distance from both endpoints. In coordinate geometry, you can use the midpoint formula to determine the midpoint of a segment. Let A(x 1, y 1) and B(x 2, y 2) be points in a coordinate plane. The midpoint of AB is x 1 + x 2 , y 1 + y 2 . 2 2 The slope is the measure of the incline of the line. If the slope is positive, the line goes up when moving from left to right, and if the slope is negative, the line goes down when moving from left to right. The letter m is often used to represent the slope of a line. Properties of slope include the following.

(

)



1.

rise y – y1 The slope, m, of a line that contains (x 1, y 1) and (x 2, y 2) is m = = 2 . run x – x 2 1



2.

Two non-vertical lines are parallel if and only if they have the same slope.



3.

Two non-vertical lines with slopes of m1 and m2 are perpendicular if and only if m2 is the negative reciprocal of m1. That is m2 = - 1 . m1 This also means m1 • m2 = -1.

Equations can be changed to the slope-intercept form by solving the equation for y. In this case, the form y = mx + b is used where m is the slope of the line and b tells where the equation intercepts the y-axis. The slope-intercept form of the equation of a line can be graphed on a coordinate plane. Express the equation in terms of a single y, that is y = mx + b. Remember that b is the yintercept or where the line crosses the y-axis, so mark the number for the b term on the y-axis. If there is no term for b, then the intercept is 0. The number in front of the x (represented in this form by m) is the slope. To graph the line, the slope must be written as a fraction. If it is already m a fraction, leave it. If it is a whole number, put the whole number over 1, that is 1 . Start at

( )

the y-intercept and move parallel to the x-axis the number of spaces in the denominator. If the fraction is positive, move up the number of spaces in the numerator. If it is negative, move down the number of spaces in the numerator. Mark this point. Draw a line to connect the y-intercept with the new point. The equation of a line can be found if you know the slope and a point on the line. Use the form for finding the intercept y = mx + b. Substitute the slope and the coordinates for the point and solve for b. Then substitute m and b into the equation y = mx + b. The equation for a line can also be found using the two-point method. Using this method, find the slope of the line by finding the change in x and the change in y. Find the © Mark Twain Media, Inc., Publishers

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Chapter 2: Coordinate and Non-coordinate Geometry

Geometry Practice

Name:

Date:

Chapter 2: Coordinate and Non-coordinate Geometry (cont.) change by subtracting one pair of coordinates from the other. Find the slope by dividing the change in y by the change in x. Substitute the slope into the form y = mx + b. Substitute one pair of coordinates into the equation and solve for b. Substitute the slope and b into the equation to find the equation of the line. The length of a line segment is the distance between the two endpoints of the segment. Distance is measured in linear measurements like meters, centimeters, inches, feet, etc. To find the distance, using the endpoints, subtract the smaller x-coordinate from the larger xcoordinate and square the difference. Then subtract the smaller y-coordinate from the larger y-coordinate and square the difference. Next, add the two squared differences and find the square root. The square root is the distance from one endpoint to the other. The formula for finding the distance between any two points (x 1, y 1) and (x 2, y 2) is (x 2 – x 1)2 + (y 2 – y 1)2 . Examples of Midpoint Formulas, Slope, Equations, Distance Example of Finding the Midpoint of a Segment: y 6 5

M

4 3 2 1

0 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 -1 -2 -3

D



M = (7, 4) D = (1, -2) Let M = (x1, y1) and D = (x2, y2)



x + x2 y1 + y2 The midpoint of DM = 1 , 2 2 7 + 1 , 4 + -2 2 2







The midpoint of DM is at (4, 1).

( )

( ( *s , @s ) = (4, 1)

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22

)

x

Chapter 2: Coordinate and Non-coordinate Geometry

Geometry Practice

Name:

Date:

Chapter 2: Coordinate and Non-coordinate Geometry (cont.) Example of Finding the Slope of a Line: y A

7 6 5

M

4 3 2 1

0 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 12 -1

x



The line on the graph above contains M = (7, 4) and A = (-2, 7).



y – y1 m = rise = 2 run x2 – x1



m = rise = 7 – 4 run -2 – 7



7 – 4 = 3 = - 1 -2 – 7 -9 3

!d

The slope of the line is - .

© Mark Twain Media, Inc., Publishers

23

Chapter 2: Coordinate and Non-coordinate Geometry

Geometry Practice

Name:

Date:

Chapter 2: Coordinate and Non-coordinate Geometry (cont.) Example of Identifying Parallel Lines:

Two non-vertical lines are parallel if and only if they have the same slope.

11 10 9 8 7 6 5 4 3 2 1 0 -11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 -11

A



M = (7, 4) and A = (-2, 7). We found the slope of that line to be - .



B = (-7, -4) and C = (2, -7)



m = rise = -7 – -4 run 2 – -7



-7 – -4 = -3 = - 1 2 – -7 9 3



The slope of the line is - . Two lines have the same slope, so they are parallel.

!d

M

B

C

!d

Example of Identifying Perpendicular Lines:

y 7

Two lines with slopes of m1 and m2 are perpendicular if and only if m2 is the negative reciprocal of m1. That means m1 • m2 = -1.

A

6

4 3

D

C

5

B

2 1

0 -4 -3 -2 -1 0 1 2 3 4 5 6 7 -1

Consider AB where A = (-2, 5) and B = (2, 3). y – y1 m = rise = 2 run x2 – x1



Consider DC where D = (-1, 2) and C = (1, 6).

















y – y1 m = rise = 2 run x2 – x1







m = 6 – 2 = 4 = 2 1 – -1 2

m = 3 – 5 = -2 = - 1 2 – -2 4 2 © Mark Twain Media, Inc., Publishers

24

x

Chapter 2: Coordinate and Non-coordinate Geometry

Geometry Practice

Name:

Date:

Chapter 2: Coordinate and Non-coordinate Geometry (cont.)

!s

The slope of AB is - . -





The slope of DC is 2.

!s • 2 = - !s • @a = - @s = -1, so the lines are perpendicular.

Example of the Slope-Intercept Method: 2x + y – 2 = 0 Solve for y. 2x + y – 2 – 2x = 0 – 2x y – 2 = -2x y – 2 + 2 = -2x + 2 y = -2x + 2 The slope of this line = -2. The y-intercept is 2, so the line intercepts the y-axis at point (0, 2). Example of Graphing Using the Intercept Method: 2y – 4x = 8 2y – 4x + 4x = 8 + 4x 2y = 4x + 8 2y = 4x + 8 2 2

y 7 6 5 4

y = 2x + 4

3 2 1 0 -4 -3 -2 -1 0 1 2 3 4 5 6 7 -1

x

Using the form y = mx + b, 4 is the y-intercept, and 2 is the slope. Mark the point at (0, 4). Since 2 is an integer, think of the slope as the fraction @a. The denominator is 1, so move right one space. The numerator is 2, and the fraction is positive, so move up 2 spaces to the point (1, 6). Mark this point, and then draw a line connecting the two points.

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Chapter 2: Coordinate and Non-coordinate Geometry

Geometry Practice

Name:

Date:

Chapter 2: Coordinate and Non-coordinate Geometry (cont.)

Example of Finding the Equation of a Line Using Slope-Intercept: The slope is -5 and one point is (-2, -1). y = mx + b -1 = -5(-2) + b -1 = 10 + b -1 – 10 = 10 + b – 10 -11 = b y = -5x -11 Example of Finding the Equation of a Line Using Two Points: Points (3, 4) and (6, 2) Change in y = 4 – 2 = 2 Change in x = 3 – 6 = -3 Divide the change in y by the change in x. -

@d is the slope

y = mx + b

@d

4 = - (3) + b 4 = -2 + b 4 + 2 = -2 + b + 2 b=6 So the equation for the line is y = - x + 6 or 3y = -2x + 18.

@d

Example of Finding the Distance: Find the distance between (5, 5) and (1, 2). (5 – 1)2 + (5 – 2)2 = (4)2 + (3)2 = 16 + 9 = 25 = 5 So the distance between the two points is 5 units.

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Geometry Practice

Chapter 2: Coordinate and Non-coordinate Geometry

Name:

Date:

Chapter 2: Coordinate and Non-coordinate Geometry (cont.) Practice: Midpoint Formulas, Slope, Equations, Distance Directions: Find the midpoints of the lines in problems 1–4 using the given endpoints. Check by graphing the results on your own graph paper. 1.

A = (5, 3) and B = (-1, 5)

2.

C = (-4, -4) and D = (6, 6)

3.

E = (2, -3) and F = (4, 11)

4.

G = (3, 9) and H = (3, 3)

Directions: Use the formula for slope to solve problems 5–10. Use a coordinate plane to help solve these problems. 5.

Find the slope between A (2, 3) and B (-1, 4).





6.





11 10 9 8 7 6 5 4 3 2 1 0 -11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 -11

11 10 9 8 7 6 5 4 3 2 1 0 -11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 -11

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Find the slope between C (4, 2) and D (1, 5).

27

Chapter 2: Coordinate and Non-coordinate Geometry

Geometry Practice

Name:

Date:

Chapter 2: Coordinate and Non-coordinate Geometry (cont.) 7.

Find the slope between E (-3, 4) and F (-2, 6). 11 10 9 8 7 6 5 4 3 2 1 0 -11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 -11

Directions: For questions 8–10 find the slope for each of the pairs of lines. Draw the lines on the coordinate plane. Explain which are parallel and which are perpendicular. 8.

AB goes through A (-2, 1) and B (1, 2) 9. EF goes through E (0, -3) and F (1, 0) CD goes through C (-2, 2) and D (-1, -1) GH goes through G (0, 3) and H (3, 1)







11 10 9 8 7 6 5 4 3 2 1 0 -11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 -11

© Mark Twain Media, Inc., Publishers

11 10 9 8 7 6 5 4 3 2 1 0 -11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 -11

28

Chapter 2: Coordinate and Non-coordinate Geometry

Geometry Practice

Name:

Date:

Chapter 2: Coordinate and Non-coordinate Geometry (cont.) 10.

IJ goes through I (0, -4) and J (3, -3) KL goes through K (0, 1) and L (3, 2)

11 10 9 8 7 6 5 4 3 2 1 0 -11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 -11



Directions: Using the slope-intercept form, find the slope and graph the line. 11.

3x – y + 2 = 0







12.

11 10 9 8 7 6 5 4 3 2 1 0 -11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 -11

© Mark Twain Media, Inc., Publishers

5x + 2y = -8 11 10 9 8 7 6 5 4 3 2 1 0 -11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 -11

29

Chapter 2: Coordinate and Non-coordinate Geometry

Geometry Practice

Name:

Date:

Chapter 2: Coordinate and Non-coordinate Geometry (cont.) 13.

y + 4 – 4x = 0 11 10 9 8 7 6 5 4 3 2 1 0 -11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 -11

Directions: Find the equation of the line using the slope-intercept method. 14.

Slope = 3, Point (4, -4)



15.

Slope = !f, Point (0, -1)





Directions: Find the equation of the line using the two-point method. 16.

(2, 3) and (6, 2)

17.

(1, 6) and (2, 1)

18. (6, -1) and (4, -2) Directions: Find the distance between two points. 19.

(3, 2) and (1, 8)

20.

(-3, 2) and (4, -1)

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Chapter 2: Coordinate and Non-coordinate Geometry

Geometry Practice

Name:

Date:

Chapter 2: Coordinate and Non-coordinate Geometry (cont.) Challenge Problems: Midpoint Formulas, Slope, Equations, Distance Directions: For questions 1–4, match the equations with the graphs. State whether the lines are parallel, perpendicular, or neither. 1.

y – 3x = 1



y + 3x = -2









A.

y 6 4



2

2.

y = 3x – 3



y = -!dx + 2







(0, 2)

M (3, 1)

0 -6 -4 -2 -0 2 4 6



-2 (0, -2) -4



x

(3, -3)

-6

3.

y + 4 = !dx



y – 1 = !dx







y





4.

y = -!dx – 2



y = -!dx + 2







B.



6 4 (0, 1) 2

(3, 2)

0 -6 -4 -2 -0 2 4 6



-2 (0, -4) -4

x

(3, -3)

-6

C.





y







D.

y

6

6

4

4

(0, 2)2

2

(3, 1)

0 -6 -4 -2 -0 2 4 6 (1, 0) -2 (0, -3) -4

x

(0, 1)

0 -6 -4 -2 -0 2 4 6 (0, -2) -2 -4

-6

© Mark Twain Media, Inc., Publishers

(1, 4)

-6

31

(1, -5)

x

Chapter 2: Coordinate and Non-coordinate Geometry

Geometry Practice

Name:

Date:

Chapter 2: Coordinate and Non-coordinate Geometry (cont.) Checking Progress: Coordinate Geometry 1.

Find the midpoint of BC where B = (6, 3) and C = (1, -2)

2.

Find the midpoint of AD where A = (3, -2) and D = (5, 12)

For questions 3–4, assume D is the midpoint of AE. 3.

AD has a length of 5. What is the length of DE?

4.

DE has a length of 11. What is the length of AE?

5.

Find the slope FG where F = (-2, 5) and G = (2, 3)

6.

Find the slope. HI where H = (1, 6) and I = (-1, 2)

7.

Are the lines in questions 5 (FG) and 6 (HI) parallel, perpendicular, or neither? Explain.

Graph the lines for 8–10 on your own graph paper. 8.

Find the slope of the line and graph. 4y – 8x = 16

9.

Find the equation of the line and graph. Slope is -7, and one point is (-3, 0)

10.

Find the equation of the line and graph. Points (4, 5) and (6, 7)

11.

Find the distance between points A and C between (7, 7) and (1, 4).

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Chapter 3: Angles

Geometry Practice

Name:

Date:

Chapter 3: Angles Basic Overview: Measuring and Classifying Angles In the figure below, we have two rays AB and AC. The angle could be labeled angle BAC or CAB. This can also be written as ∠BAC or ∠CAB. An angle can also be labeled using the letter at the vertex. So the angle could just be labeled ∠A. An angle can also be labeled using a number inside its vertex. When using a number to label the angle, make a small arc inside the angle. So the angle could simply be labeled ∠1.

B a

1

C Angles can be measured using a protractor. To measure an angle using a protractor, place the center mark on the straight edge of the protractor over the vertex of the angle. Place the horizontal line of the protractor along one side of the angle. Read the number where the other ray crosses or intersects the protractor. This is the measure of the angle. Sometimes the angles may be greater than 180 degrees. Angles that are greater than 180 degrees are shown by adding a small arc to the angle. To get the measure of the angle, subtract the small angle from 360 degrees. Acute angles are less than 90 degrees. Right angles measure exactly 90 degrees. Obtuse angles are more than 90 degrees but less than 180 degrees. Straight angles are exactly 180 degrees. Reflex angles are greater than 180 degrees but less than 360 degrees. Two angles are vertical if they are non-adjacent angles formed by a pair of intersecting lines. Two angles are complementary if the two angles add up to 90 degrees, and they are supplementary if the two angles add up to 180 degrees and form a straight line. Two angles are congruent if they have the same measure. Adjacent angles share a common vertex and side but have no common interior points. An angle bisector is a ray that divides an angle into two equal parts. Two adjacent angles form a linear pair if their non-common sides are on the same line; hence, a linear pair is supplementary in both placement and measure.

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Geometry Practice

Chapter 3: Angles

Name:

Date:

Chapter 3: Angles (cont.) Examples of Measuring and Classifying Angles Example of Using a Protractor to Measure An Angle Greater Than 180 Degrees:

F

G

H

This angle has an arc indicating the angle is greater than 180 degrees. The measure of the small angle is 20 degrees. Subtract the small angle from 360 degrees. m∠FGH = 360° – 20°. The difference is 340 degrees. m∠FGH = 340°. Examples of Acute Angles:





15°

70°

45°

These angles are all less than 90 degrees, so they are acute angles.

Examples of Right Angles: 90°

90°



These angles are right angles because they are both 90-degree angles.

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Geometry Practice

Chapter 3: Angles

Name:

Date:

Chapter 3: Angles (cont.) Examples of Obtuse Angles:

150° 110°





These angles are both obtuse because they are more than 90 degrees but less than 180 degrees.

Example of a Straight Angle: 180°

This is a straight angle because it is exactly 180 degrees.

Examples of Reflex Angles: 280°

325°



These are reflex angles because they are greater than 180 degrees but less than 360 degrees.

Examples of Vertical Angles: 1 4

© Mark Twain Media, Inc., Publishers

35

3

2

Chapter 3: Angles

Geometry Practice

Name:

Date:

Chapter 3: Angles (cont.)



Angles 1 and 3 are vertical angles because their sides form two pairs of opposite rays. Angles 2 and 4 are vertical angles because their sides also form two pairs of opposite rays.

Example of Complementary Angles:

A D 40° 50° B



C

∠ABD is 40 degrees, and ∠DBC complementary angles because their sum is equal to 90 degrees.

is

50

degrees.

Example of Supplementary Angles:

∠ABD

and

∠DBC

are

D

150° 30°

A B ∠ABD and ∠DBC are supplementary because a sum of 180°. These two angles also form a linear pair of angles.

they

form

a

C straight

line

and

have

D

Examples of Congruent Angles:

160°

A

B

20°

20°

C 160°

E

∠ABD and ∠EBC both have the same measure of 160°, so they are congruent. ∠ABE and ∠DBC both have the same measure of congruent.

© Mark Twain Media, Inc., Publishers

36

20°,

so

they

are

also

Chapter 3: Angles

Geometry Practice

Name:

Date:

Chapter 3: Angles (cont.) Example of an Angle Bisector: A D 40° 40°

B

C The angle bisector is BD because it divides ∠ABC into two equal parts. Example of Adjacent Angles: A D 40° B

∠ABD is adjacent to have any common interior points.

40° ∠DBC

because

C they

share

the

common

side

BD

and

do

not

Example of Linear Pairs:

B

A

D

C

∠ADB and ∠BDC form a linear pair because they are adjacent angles and the non-common sides are on the same line. Note that they are supplementary, also.

© Mark Twain Media, Inc., Publishers

37

Chapter 3: Angles

Geometry Practice

Name:

Date:

Chapter 3: Angles (cont.) Practice: Measuring and Classifying Angles Directions: Measure the angles. 1.

m∠ =











2.

m∠ =

3.

m∠ =











4.

m∠ =

5.

m∠ =



6.

Draw an example of an acute angle and explain why the angle is acute.

7.

Draw an example of an obtuse angle and explain why the angle is obtuse.



© Mark Twain Media, Inc., Publishers

38

Chapter 3: Angles

Geometry Practice

Name:

Date:

Chapter 3: Angles (cont.) 8.

Draw an example of a right angle and explain why it is a right angle.

9.

Draw an example of a straight angle and explain why it is a straight angle.

10.

Draw an example of a reflex angle and explain why it is a reflex angle.

11.

Draw an example of complementary angles and explain why they are complementary.

12.

Draw an example of adjacent angles and explain why they are adjacent angles.

13.

Draw an example of a linear pair and explain why it is a linear pair.

14.

Draw an example of congruent angles and explain why they are considered congruent.

15.

Draw an example of a vertical angle and explain why it is a vertical angle.

© Mark Twain Media, Inc., Publishers

39

Geometry Practice

Chapter 3: Angles

Name:

Date:

Chapter 3: Angles (cont.) Challenge Problems: Measuring and Classifying Angles 1.

Which angle is angle G? E G

F H

2.

Identify the points in the interior of the angle, exterior of the angle, and on the angle itself.

A C B D 3.

Describe four different ways the angle below can be labeled.

G

F

3 H

4.

Identify the adjacent angles, complementary pair, supplementary pair, the congruent pair, vertical angles, linear pairs, and the angle bisector. Explain why the angles are adjacent, complementary, supplementary, congruent, vertical, or the angle bisector.



A.



1 4

© Mark Twain Media, Inc., Publishers

40

3

2

Geometry Practice

Chapter 3: Angles

Name:

Date:

Chapter 3: Angles (cont.) B.

B

A

C D

C. A D C

B



D.





A D

B

© Mark Twain Media, Inc., Publishers

40° 40°

41

C

Geometry Practice

Chapter 3: Angles

Name:

Date:

Chapter 3: Angles (cont.)

E.



D

A

C B

E

F.





D

A

C B



G.



A D

B

© Mark Twain Media, Inc., Publishers

42

C

Geometry Practice

Chapter 3: Angles

Name:

Date:

Chapter 3: Angles (cont.) Checking Progress: Measuring and Classifying Angles Directions: Match the words with the definitions. 1. 2. 3. 4.

An angle that measures 90 degrees The sum of the two angles is 180 degrees. Two angles with exactly the same measure An angle with a measure of less than 90 degrees

5.

The sum of two angles is 90 degrees.

6.

The angles formed when two lines intersect

7.

An angle that is greater than 90 degrees

8.

An angle with a measure of greater than 180 but less than 360 degrees

9.

A ray that divides an angle into two congruent angles

10.

An angle that measures exactly 180 degrees

11.

Adjacent angles and the noncommon sides are opposite rays.

© Mark Twain Media, Inc., Publishers

43

A.

acute angle

B.

angle bisector

C.

complementary angles

D.

congruent angles

E.

linear pair

F.

obtuse angle

G.

reflex angle

H.

right angle

I.

straight angle

J.

supplementary angles

K.

vertical angles

Chapter 4: Patterns and Reasoning

Geometry Practice

Name:

Date:

Chapter 4: Patterns and Reasoning Basic Overview: Patterns, Structure of Geometry, Conditional Statements, and Deductive Reasoning Reasoning in geometry usually consists of three parts. One part is using inductive reasoning to look for patterns. Inductive reasoning is making a conjecture about several examples after looking for a pattern.The second part is making conjectures. Making conjectures is drawing inferences based upon empirical evidence. The third part of reasoning in geometry is verifying the conjectures made using logical reasoning. A pattern has three characteristics: a unit, repetition, and a system of organization. Geometry is guided by the rules of geometry. To prove a new rule, you must use at least one other rule. Some rules for the study of geometry called postulates or axioms must be accepted without proving them to be true. Once you know the postulates, you can develop new rules called theorems. A conditional statement or if-then statement has two parts: the hypothesis and the conclusion. The conditional statement claims: if the hypothesis (p) is true, then the conclusion (q) is true. In other words, if p then q, which can be symbolized as: p ⇒ q. By exchanging the hypothesis and conclusion, the converse of a conditional statement is formed. The converse of a conditional statement changes “if p then q” to “if q then p,” which can by symbolized as: q ⇒ p . Bi-conditional statements are in the form: “p if and only if q,” which can be symbolized as: p ⇔ q . A conditional statement and its converse do not need to be true at the same time, but if they are, then together they form a bi-conditional statement. Geometry uses conjectures or inferences based on empirical evidence to create new theorems. Once the theorems are proposed from the conjectures, they are verified using logical or deductive reasoning. Examples of Patterns and Reasoning Number Systems:

Counting is based on patterns of the decimal number system, which is based on 10 digits. {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}

Binary Codes Used in Computers: Binary Number 24 23

22

21

20 Decimal Equivalent



1 1 1

1 1 0 1

0 1 1 0





10 111 10101 11110

© Mark Twain Media, Inc., Publishers

1 1

0 1

44



2 7 21 30

Chapter 4: Patterns and Reasoning

Geometry Practice

Name:

Date:

Chapter 4: Patterns and Reasoning (cont.) Example of Postulate One Using an Inch Scale:

AB = |6 – 3| = 3 in.

Postulate One: The points on a line can be matched, one-to-one, with the set of real numbers. A real number that corresponds with a point is the location of that point. The distance AB between the two points A and B on a line is equal to the absolute value of the difference between the locations of A and B. A

B

Example of the Vertical Angles Theorem:

Proof: Use the figure below.

1 4

3 2

Given: ∠1 and ∠2 are vertical angles. Prove ∠1 @ ∠2 Because ∠1 and ∠3 form a linear pair, they are supplementary angles or angles the sum of which is 180°. The measure of ∠1 + measure of ∠3 = 180°. Because ∠2 and ∠3 form a linear pair, they are supplementary angles. Because ∠1 and ∠2 are each supplementary to ∠3, they must be congruent (based on the Congruent Supplements Theorem of Supplementary Angles).

© Mark Twain Media, Inc., Publishers

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Chapter 4: Patterns and Reasoning

Geometry Practice

Name:

Date:

Chapter 4: Patterns and Reasoning (cont.) Practice: Patterns, Structure of Geometry, Conditional Statements, and Deductive Reasoning Directions: Complete the exercises as directed in each section. Find the patterns. 1.

How many non-reflex angles do 5 rays with a common endpoint make? 6 rays?





#of Rays





# of non-reflex Angles



Using the completed table as the basis for conjecture, predict the number of non-reflex





2

3

4

5

1

3

6



6

angles that 12 rays with a common endpoint will generate. 2.

What is the next figure in this sequence? Draw it.

3.

What is the next number in this sequence?





0, 4, 8, 12

For problems 4–6, identify the hypothesis, conclusion, and converse statement. 4.

If someone will pay $25 for the table, then I will sell it.

5.

If you are willing, then you are able.



© Mark Twain Media, Inc., Publishers

46

Chapter 4: Patterns and Reasoning

Geometry Practice

Name:

Date:

Chapter 4: Patterns and Reasoning (cont.) 6.

If two angles have the same measurement, then they are congruent.



Write the statements below in the bi-conditional form. 7.

Two angles that share a common side and have no common interior points are adjacent.

8.

A bisector ray divides an angle into two equal parts.



Decide if the statement is true or false and explain your reason. 9.

A point may lie on more than one line.

10.

If x 2 = 4 then x = 2

11.

Complete the following statement to make the statement true:



x 2 = 9 if and only if

© Mark Twain Media, Inc., Publishers

47

Chapter 4: Patterns and Reasoning

Geometry Practice

Name:

Date:

Chapter 4: Patterns and Reasoning (cont.) Challenge Problems: Patterns, Structure of Geometry, Conditional Statements, and Deductive Reasoning 1.

Complete the table below, construct a scatter plot where a∠1 = x and a∠2 = y to represent the data, and write an equation to represent the relationship.





a∠1 5°

15°



a∠2 85°

75°



25°

35°

45°

55°

65°

75°

85°

y 85 80 75 70 65 60 55 50 45 40 35 30 25 20 15 10 5 0 -10 -5 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 -5

© Mark Twain Media, Inc., Publishers

48

x

Geometry Practice

Chapter 4: Patterns and Reasoning

Name:

Date:

Chapter 4: Patterns and Reasoning (cont.) 2.

Develop a proof.













D

A

C B



∠ABD and ∠DBC form a linear pair.



∠ABD = 140°



Prove that angle ∠DBC = 40°.

3.

The angle addition postulate states if B is in the interior of ∠AOC, then m∠AOB + m∠BOC = m∠AOC.

B

A

C O

m∠AOC = 100°, ∠AOB = 60°



Find m∠BOC.

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49

Geometry Practice

Chapter 4: Patterns and Reasoning

Name:

Date:

Chapter 4: Patterns and Reasoning (cont.) Checking Progress: Patterns, Structure of Geometry, Conditional Statements, and Deductive Reasoning 1.

What are the three characteristics of a pattern?

2.

Give two examples of a pattern.

Fill in the blanks. 3.

or axioms are statements accepted as true without proof.

4.

A

is a statement that must be proven true.

5.

A

statement is an if-then statement.

6.

statements are in the form of if and only if.

7.

reasoning is using the laws of logic to prove statements from known statements.

8.

reasoning means making a conjecture about several examples after looking at a pattern.

9.

Making

means drawing inferences based upon

empirical evidence. 10.

Conjectures made can be verified using

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50

reasoning.

Geometry Practice

Chapter 5: Triangles

Name:

Date:

Chapter 5: Triangles Basic Overview: Parts of a Triangle, Similar and Congruent Triangles, Properties of Triangles, Area and Perimeter The interior angles of a triangle are each less than 180 degrees. When a side of a triangle is extended past a vertex, the angle formed is called an exterior angle. This exterior angle forms a linear pair with its adjacent interior angle. Theorems for Interior Angles of Triangles • Each angle of a triangle has a measure greater than 0 degrees and less than 180 degrees. • The sum of the measures of the angles of a triangle is 180 degrees. Theorems for Exterior Angles of Triangles • The measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles. • The measure of any exterior angle of a triangle is greater than the measure of either of the two non-adjacent interior angles. Triangles have three sides and three angles. There are special names given to the angles and sides of the triangle based on their relationships among each other. An acute triangle is a triangle in which all of the interior angles are less than 90 degrees. This means that every angle of an acute triangle is an acute angle. An obtuse triangle is a triangle with one obtuse angle. Remember that an obtuse angle is greater than 90 degrees. A right triangle is a triangle with one right angle, and a right angle is 90 degrees. An isosceles triangle is a triangle with two congruent angles and two congruent sides. An equilateral triangle is a triangle with three equal sides and with three congruent or equal angles. A triangle with three equal angles is also called an equiangular triangle. A scalene triangle is a triangle with no congruent sides or angles. This means all of the sides and angles have different measures. A scalene triangle can be an acute, obtuse, or right triangle. Similar triangles have these characteristics: the three angles are congruent or equal, respectively, to the three corresponding angles of the other triangle; and the ratio of two corresponding sides of one triangle is equal to the ratio of the two corresponding sides of the other triangle. Congruent triangles have the same size and shape, which means the angles and sides of one triangle are congruent to the corresponding angles and sides of the other triangle. The Pythagorean Theorem states: the square of the length of one of the legs (s1) plus the square of the length of the second leg (s2) is equal to the square of the length of the hypotenuse (h). So the equation could be represented by s12 + s22 = h 2. The Mid-segment Theorem states: the segment connecting the midpoints of two sides of a triangle is parallel to the third side and is half its length.

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Chapter 5: Triangles

Geometry Practice

Name:

Date:

Chapter 5: Triangles (cont.) The triangle inequality theorem states that the sum of the lengths of any two sides (s1 and s2) of a triangle is greater than the length of the third side (s3). An angle bisector is a ray that divides an angle into two equal angles. A perpendicular bisector of a segment is a line that is perpendicular to the segment at its midpoint. A median is a segment whose endpoints are a vertex and the midpoint of the opposite side. An altitude is a segment from a vertex that is perpendicular to the opposite side or to the line containing the opposite side. The perimeter of a triangle is the distance around the triangle, p = s1 + s2 + s3. The area of a triangle can be found by multiplying half of the base times the altitude, A = !s(b • h). Examples of Parts of a Triangle, Similar and Congruent Triangles, Properties of Triangles, Area and Perimeter Example of an Interior Angle: The interior angles are the angles at the vertices ∠ABC, ∠BCA, and ∠CAB. A

B

C

Example of an Exterior Angle: ∠2 is an exterior angle.

A Figure 1

B Example of Similar Triangles:

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52

2

1

C

Chapter 5: Triangles

Geometry Practice

Name:

Date:

Chapter 5: Triangles (cont.) Example of Congruent Triangles:

Examples of the Pythagorean Theorem: Find the hypotenuse of the right triangle if you know the length of the legs. s1 = 2 cm s2 = 3 cm Hypotenuse s12 + s22 = h 2 s1 (2)2 + (3)2 = h 2 h 2 4+9=h 13 = h 2 s2 13 = h 2 Legs 13 = h Find the leg of the right triangle if you know the length of one leg and the hypotenuse. s2 = 6 cm h = 10 cm s1 = ? s12 + s22 = h 2 s12 + (6)2 = (10)2 s12 + 36 = 100 s12 + 36 – 36 = 100 – 36 s12 = 64 s12 = 64 s1 = 8 cm

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Chapter 5: Triangles

Geometry Practice

Name:

Date:

Chapter 5: Triangles (cont.) Example of Finding the Mid-Segments of a Triangle:

Find the midpoint of BC labeled D and Midpoint of AC labeled E on the drawing above. Midpoint formula:

x + x y + y , ( 2 2 ) 1

2

1

2

(



)

(

Midpoint of BC = 4 + 6 , 5 + (-1) 2 2

Midpoint of AC = 6 + (-2) , -1 + 3 2 2

Midpoint of BC = ( Aw:, $s ) = (5, 2)

Midpoint of AC = ( $s, @s ) = (2, 1)

Coordinates of the midpoint D are (5, 2).

Coordinates of the midpoint E are (2, 1).



Example of Mid-segment Theorem of a Triangle: Using the midpoints in triangle ABC above. Draw segment ED.

Find the slope of AB and ED. y – y1 Formula for slope: m = rise = 2 run x2 – x1 © Mark Twain Media, Inc., Publishers

54

)

Chapter 5: Triangles

Geometry Practice

Name:

Date:

Chapter 5: Triangles (cont.) AB m = 5 – 3 4 – (-2)







ED m = 2 – 1 5 – 2

AB m = @h = !d ED m = !d The slopes of AB and ED are the same, so the lines are parallel. Example of a Perpendicular Bisector of a Triangle: Find the midpoint of AC labeled D on the drawing below.

(

Midpoint of AC = 0 + 10 , 0 + 0 2 2

)

Midpoint of AC = (Aw: , )s ) = (5, 0) The midpoint D has the coordinates of (5, 0). Using a protractor, draw a perpendicular line from the point to line AC.

(10,0) (5,0)

Example of Finding the Perimeter of a Triangle: AB = 1.5 cm, BC = 5 cm, CA = 4 cm p = s1 + s2 + s3 p = 1.5 + 5 + 4





p = 10.5 cm The perimeter is 10.5 cm.

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Geometry Practice

Chapter 5: Triangles

Name:

Date:

Chapter 5: Triangles (cont.) Example of Finding the Area of a Triangle:

Each square on the grid represents one centimeter. Base = 16 cm and altitude = 10 cm A = !s(bh) A = !s(16 • 10) A = !s(160) A = 80 Add the units. A = 80 cm2

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Geometry Practice

Chapter 5: Triangles

Name:

Date:

Chapter 5: Triangles (cont.) Practice: Parts of a Triangle, Similar and Congruent Triangles, Properties of Triangles, Area and Perimeter Directions: State whether the following statements are always [A], sometimes [S], or never [N] true. 1.

An equilateral triangle is equiangular.

2.

A right triangle is scalene.

3.

An isosceles triangle is a right triangle.

4.

A scalene triangle is isosceles.

5.

An obtuse triangle is scalene.

6.

A right triangle is obtuse.

7.

An acute triangle is equiangular.

8.

An equilateral triangle is acute.

9.

The sum of the measures of the two smaller angles in a right triangle is 90°.

10.

A triangle has a line of symmetry.

11.

An equilateral triangle has three lines of symmetry.

12.

A scalene triangle has a line of symmetry.

13.

If three angles in one triangle are equal to three angles in another triangle, the two triangles are congruent.

14.

A side of a triangle is also an altitude of the triangle.

15.

If three sides of one triangle are congruent to three sides of a second triangle, the triangles are congruent.

Directions: State whether the following statements are true [T] or false[F]. 16.

A triangle can be made with one angle having a measure of 179°.

17.

The sum found when adding the measure of three exterior angles, one at each vertex in a triangle, is 360°.

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Geometry Practice

Chapter 5: Triangles

Name:

Date:

Chapter 5: Triangles (cont.) 18.

The longest side in a triangle is called the hypotenuse.

19.

The segment connecting the midpoints of two sides of a triangle is parallel to the third side.

20.

An exterior angle is greater than either non-adjacent interior angle.

Directions: For problems 21–24: In right triangle ABC, with B being the right angle, answer the following. 21.

If AB = 5 cm, BC = 12 cm, what is the length of AC ?

22.

If AB = 5 cm, and AC = 12 cm, what is the length of BC?

23.

If AB = 5 cm and AB = BC, what is the length of AC ?

24.

If AB = BC and AC = 5 cm, what is the length of AB?

25.

What is the length of the altitude in an equilateral triangle with sides 10 cm long?



26.

Find the midpoint of the segment with endpoints (2, 7) and (18, 9).

27.

If (3, 5) is the midpoint between (2, 7) and (a, b), what are the values for a and b ?



28.

If (3, b) is the midpoint between (a, 7) and (13, 21), what are the values for a and b?



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Geometry Practice

Chapter 5: Triangles

Name:

Date:

Chapter 5: Triangles (cont.) 29.

If (6, -5) is the midpoint between (1, 9) and (a, b), what are the values of a and b?

30.

Consider triangle ABC, where A, B, and C have coordinates (12, 66), (18, 66), and (18, 26) respectively.



a.

What is the area of this triangle?



b.

What is the distance from the midpoint of BC to the midpoint of AC ?







c.







d.

What is the distance from the midpoint of AC to the midpoint of AB?

What is the area of the triangle formed by joining the three midpoints of triangle ABC?



e.

What is the length of the segment joining the midpoints of AB and BC?





31.

Find the perimeter of an equilateral triangle with a side of the length c.

32.

Find the perimeter and area of a right triangle with legs of 6 cm and 8 cm.



Perimeter:

33.

Consider two non-congruent triangles with the base length of 8 and altitude of 5. What



Area:

is the area of each?

34.

Find the perimeter and area of a right triangle with one leg 20 cm and the hypotenuse 25 cm. Perimeter:

35.

Area:

What is the perimeter of an isosceles right triangle with legs of the length 12 cm?



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Chapter 5: Triangles

Geometry Practice

Name:

Date:

Chapter 5: Triangles (cont.) Challenge Problems: Parts of a Triangle, Similar and Congruent Triangles, Properties of Triangles, Area and Perimeter 1.

Is the following statement always [A], sometimes [S], or never [N] true?



Two equilateral triangles are similar.

2.

If possible, draw a triangle that is isosceles and obtuse.

3.

If possible, draw a triangle that is acute and right.

4.

If possible, draw a triangle that is right and obtuse.

5.

If possible, draw a right triangle with exactly one line of symmetry.

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Chapter 5: Triangles

Geometry Practice

Name:

Date:

Chapter 5: Triangles (cont.) Directions: Circle the correct answer for questions 6–11. 6.

T or F? A triangle can be made with sides of the lengths 5 cm, 8 cm, and 15 cm.

7.

T or F? If you connect the midpoints of the three sides of a triangle to form another triangle, the two triangles are similar.

8.

T or F? If two sides and an angle of one triangle are congruent to two sides and an angle of a second triangle, the triangles are congruent.

9.

T or F? Two triangles can have the same area but not the same perimeter.

10.

T or F? The three angle bisectors intersect in a single point.

11.

T or F? A median of a triangle divides the triangle into two smaller triangles of equal area.

12.

In right triangle ABC, with B being the right angle, if AB = 15 cm and AC = 125 cm, what is the length of CB?

13.

Give the lengths of the sides of a triangle that have integer sides and perimeter of 9.



14.

The sides of a triangle measure 6 cm, 8 cm and x cm in length. If x can only be measured for an integer number of cm, then what is the smallest possible perimeter? What is the largest possible perimeter?

15.

Two triangles have the same area. One has a base of 12 cm and an altitude of x. The other has a base of 18 cm and an altitude of 10 cm. What is the value of x?

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Chapter 5: Triangles

Geometry Practice

Name:

Date:

Chapter 5: Triangles (cont.) Checking Progress: Triangles 1.

State whether the following statement is always [A], sometimes [S], or never [N] true:



An altitude of a triangle is longer than the respective base.

2.

State whether the following statement is always [A], sometimes [S], or never [N] true: Two triangles are congruent if they have the same area.

3.

T or F? If two triangles are such that two angles and a side of one are congruent to two sides and an angle of the other, the two triangles are congruent.

4.

T or F? If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.

5.

If the two legs of a right triangle are 12 cm and 7 cm, what is the length of the hypotenuse?

6.

If the two legs of a right triangle are 12 cm and 7 cm, what is the area of the triangle?

7.

If (2, 5) and (a, b ) are two points, what is the midpoint between them?

8.

If (13, b) is the midpoint between (a, 17) and (13, 21), what are the values for a and b?

9.

The sides of a triangle measure 10 cm, 15 cm and x cm in length. If x can only be measured for an integer number of cm, then what is the smallest possible perimeter? What is the largest possible perimeter?

10.

Draw and label two triangles with the same areas but different perimeters.

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62

Chapter 6: Polygons and Quadrilaterals

Geometry Practice

Name:

Date:

Chapter 6: Polygons and Quadrilaterals Basic Overview: Polygons and Quadrilaterals Polygons are plane figures that are formed by three or more segments called sides. Each side of a polygon intersects two other sides, once at each endpoint. No two sides of a polygon with a common endpoint are collinear. A polygon is convex if no line that contains a side of the polygon contains an interior point. A polygon that is not convex is called a non-convex or concave polygon. Polygons have vertices, interior angles, exterior angles, perimeter, and area. Polygons also have diagonal segments that join two nonconsecutive vertices. A polygon is equilateral if all sides are congruent. It is equiangular if all interior angles are congruent. A polygon that is equilateral and equiangular is regular, that is, all angles and sides are congruent. A diagonal of a polygon is a line segment joining non-adjacent vertices. The polygon interior angles theorem states that the sum of the measures of the interior angles of a convex n-gon (n representing the number of sides) is (n – 2) • (180°). Any convex polygon can be divided into non-overlapping triangles by drawing all the diagonals from one vertex. The number of triangles formed is always two less than the number of sides, which verifies the sum formula. All angles of a regular polygon are congruent. The corollary theorem related to polygons 1 states that the measure of each interior angle of a regular n-gon is n (n – 2)(180°). The polygon Exterior Angles Theorem states that the sum of the measures of the exterior angles, one from

each vertex of a convex polygon, is equal to 360°. The corollary to this theorem states that the 1 measure of each exterior angle of a regular n-gon is n (360°). If the lines are extended on a polygon, the interior and exterior angles are supplementary.

Quadrilaterals are four-sided polygons including squares, rectangles, trapezoids,

parallelograms, and rhombi (the plural form of rhombus). A trapezoid has one pair of parallel sides that are called bases and two non-parallel sides called legs. The median of a trapezoid is a line segment that connects the midpoints of the legs. It is also parallel to both bases. The length of the median is one-half the sum of the b + b2 two bases !s(b1 + b2) also written as 1 or the average of the two bases. Two theorems 2 © Mark Twain Media, Inc., Publishers

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Geometry Practice

Chapter 6: Polygons and Quadrilaterals

Name:

Date:

Chapter 6: Polygons and Quadrilaterals (cont.) related to trapezoids are: (1) the pair of angles adjacent to a non-parallel side of the trapezoid is supplementary, and (2) there are two such pairs of angles, since there are two non-parallel sides in a trapezoid, and the sum of all four angles of a trapezoid is 360°. Remember that two angles are supplementary if their sum is 180°. A parallelogram has four sides including two pairs of parallel sides. Theorems related to parallelograms include the following: (1) the opposite sides of a parallelogram are congruent; (2) opposite angles of a parallelogram are congruent; (3) consecutive pairs of angles of a parallelogram are supplementary; (4) the sum of the interior angles of a parallelogram is 360°; (5) each diagonal of a parallelogram separates the parallelogram into two congruent triangles; (6) the diagonals of a parallelogram bisect each other; and (7) the four non-overlapping triangles formed by the diagonals are congruent. A rhombus is also a parallelogram. It is a polygon with four equal sides. Theorems related to a rhombus include the following: (1) the diagonals of a rhombus bisect the interior angles of the rhombus; (2) the diagonals are perpendicular and separate the rhombus into four congruent triangles; (3) the diagonals of a rhombus bisect each other; (4) the opposite angles in a rhombus are congruent; and (5) any consecutive pairs of angles in a rhombus are supplementary. A square is a rectangle with four equal sides. The measure of each of the angles of a square are 90°. Related theorems include the following: (1) the measures of all of the angles in a square are equal; (2) the sum of the interior angles of a square is 360°; (3) all four sides of a square are equal; (4) the diagonals of a square are equal, bisect each other, are perpendicular to each other; and (5) the triangles formed by a diagonal are congruent.

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Geometry Practice

Chapter 6: Polygons and Quadrilaterals

Name:

Date:

Chapter 6: Polygons and Quadrilaterals (cont.) Examples of Polygons and Quadrilaterals Example of Polygon Interior Angles Theorem:





Number of Non-overlapping

Polygon

Triangle





Number of Sides (n) Triangles from one vertex 3 1

Sum of Interior Angles

1(180°) = 180°

Quadrilateral



4













2









2(180°) = 360°

Pentagon





5













3









3(180°) = 540°

Hexagon







6













4









4(180°) = 720°

Heptagon





7













5









5(180°) = 900°

Octagon







8













6









6(180°) = 1,080°

n-gon







n











n – 2







(n – 2)(180°)

Example of Corollary Theorem Related to Polygons:

Regular Polygon Number of Sides (n)

Measure of Interior Angle

1 (n – 2)(180°) n

Triangle





3





!d(3 – 2)(180°) = !d(1)(180°) = 60°

Square







4





!f(4 – 2)(180°) = !f(2)(180°) = !f(360°) = 90°

Pentagon





5





!g(5 – 2)(180°) = !g(3)(180°) = !g(540°) = 108°

Hexagon





6





!h(6 – 2)(180°) = !h(4)(180°) = !h(720°) = 120°

Heptagon





7





!j(7 – 2)(180°) = !j(5)(180°) = !j(900°) ≈ 128.6°

Octagon





8





!k(8 – 2)(180°) = !k(6)(180°) = !k(1,080°) = 135°







n





(n – 2)(180°) n

n–gon



© Mark Twain Media, Inc., Publishers

1

65

Geometry Practice

Chapter 6: Polygons and Quadrilaterals

Name:

Date:

Chapter 6: Polygons and Quadrilaterals (cont.)

Example of Finding the Measure of the Interior Angles Using Equations:





If the interior angles of a quadrilateral are put in increasing order, they increase by 30°.





What are the measures of each of the angles?





This is a quadrilateral, so we know the sum of all interior angles is 360°.





Start with ∠D since it is the smallest. ∠D = x





The next smallest angle is ∠B, so ∠B = x + 30°





m∠A = x + 30° + 30° = x + 60°





m∠C = x + 60° + 30° = x + 90°





Put the equation together:



x + (x + 30°) + (x + 60°) + (x + 90°) = 360°





Simplify:





4x + 180° = 360°





4x + 180° – 180° = 360° – 180°





4x = 180°





4x = 180° 4 4





x = 45°





Answers:

m∠D = x = 45° m∠B = x + 30° = 45° + 30° = 75° m∠A = x + 60° = 45° + 60° = 105° m∠C = x + 90° = 45° + 90° = 135°

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Geometry Practice

Chapter 6: Polygons and Quadrilaterals

Name:

Date:

Chapter 6: Polygons and Quadrilaterals (cont.)

Example of Finding the Interior or Exterior Angles of a Polygon:





m∠2 = 100° m∠8 = 40° m∠4 = m∠5 = 110°





Find the measures of ∠7, ∠3, ∠9, ∠10, ∠1, and ∠6.

Which angles are supplementary? ∠2 and ∠7, ∠8 and ∠3, ∠9 and ∠4, ∠5 and ∠10, and ∠1 and ∠6



Supplementary angles add up to 180°. Set up equations to find these measures.



m∠7 = 180° – m∠2 m∠7 = 180° – 100° m∠7 = 80°

m∠3 = 180° – m∠8 m∠9 = 180° – m∠4 m∠3 = 180° – 40° m∠9 = 180° – 110° m∠3 = 140° m∠9 = 70°







m∠10 = 180° – m∠5 m∠10 = 180° – 110° m∠10 = 70°







The sum of the interior angles of a pentagon is 540°. Subtract the measure of all of the other interior angles to find out the measure of ∠1.











m∠6 is supplementary to m∠1.

m∠1 = 540° – m∠2 – m∠3 – m∠4 – m∠5 m∠1 = 540° – 100° – 140° – 110° – 110° m∠1 = 80°

m∠6 = 180° – m∠1 m∠6 = 180° – 80° m∠6 = 100°

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Chapter 6: Polygons and Quadrilaterals

Geometry Practice

Name:

Date:

Chapter 6: Polygons and Quadrilaterals (cont.)

Example of Finding the Median of a Trapezoid:

(6, 12)

A

B

(12, 12)

(4.5, 7.5) F

E (17.5, 7.5)

C (23, 3)

D (3, 3)





Find the length of the median by finding the average of the two bases.





length of AB = b1 = 6





b + b2 length of FE = 1 = 2





6 + 20 = 26 = 13 2 2





The median of trapezoid ABCD is 13 units.





Assuming the bases are parallel to the x-axis, the median connects both legs at their midpoints.





Find the midpoints of both legs AD and BC.





x1 + x2 y1 + y2 Midpoint formula = , 2 2





AD point F = 6 + 3 , 12 + 3 2 2





AD =



length of DC = b2 = 20

(

(

((s, AwG) = (4.5, 7.5)

© Mark Twain Media, Inc., Publishers

)





68

) (

BC point E = 12 + 23 , 12 + 3 2 2 BC =

(DwG, AwG) = (17.5, 7.5)

)

Chapter 6: Polygons and Quadrilaterals

Geometry Practice

Name:

Date:

Chapter 6: Polygons and Quadrilaterals (cont.)

Examples of Trapezoid Supplementary Angles: A

B

C

D





In this trapezoid, m∠A is supplementary to m∠C, and m∠B is supplementary to m∠D.







Examples of Characteristics of a Parallelogram:

m∠A + m∠B + m∠C + m∠D = 360°

E





Opposite sides AD and BC are congruent, and DC and AB are congruent.





m∠B and m∠C are supplementary (120° + 60° = 180°). ∠A and ∠D are supplementary.





The diagonal AC forms two congruent triangles: DABC is congruent to DCDA.





The two diagonals of the parallelogram, AC and BC bisect each other at point E.

The measures of ∠B and ∠D are congruent, or are the same measure, and ∠A and ∠C are congruent.

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Chapter 6: Polygons and Quadrilaterals

Geometry Practice

Name:

Date:

Chapter 6: Polygons and Quadrilaterals (cont.)

Example of a Rhombus With Diagonals:





Because the diagonals bisect the angles of a rhombus: m∠1 and m∠2 are congruent. m∠3 and m∠4 are congruent. m∠5 and m∠6 are congruent. m∠7 and m∠8 are congruent.





The diagonals, AD and BC, are perpendicular in the rhombus above.





Each diagonal separates the rhombus into two congruent triangles.





∠A and ∠C are supplementary, or there sum is equal to 180°.





∠B and ∠D are supplementary.



Example of a Rectangle: A

C E D

B



m∠A, m∠B, m∠C, and m∠D are equal to 90°, so any two angles are supplementary.





AB and CD are parallel, and BD and AC are parallel.





Diagonal AD separates the rectangle into two congruent triangles, DACD and DABD, and the two diagonals bisect each other at point E.

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Chapter 6: Polygons and Quadrilaterals

Geometry Practice

Name:

Date:

Chapter 6: Polygons and Quadrilaterals (cont.) Practice: Polygons and Quadrilaterals Directions: In questions 1–20, state whether the following statements are always [A], sometimes [S], or never [N] true. 1.

The sum of the measures of the angles in a trapezoid is 360°.

2.

A trapezoid has two acute angles.

3.

A rectangle has exactly two lines of symmetry.

4.

A parallelogram has a line of symmetry.

5.

A rhombus has four right angles.

6.

The diagonals of a parallelogram bisect each other.

7.

A trapezoid has a line of symmetry.

8.

An equiangular quadrilateral is a square.

9.

All four sides of a rhombus are equal.

10.

The diagonals of a trapezoid are perpendicular.

11.

The diagonals of a parallelogram are perpendicular.

12.

A hexagon has an even number of lines of symmetry.

13.

A rhombus has four lines of symmetry.

14.

The sum of the measures of the exterior angles, one at each vertex, of a quadrilateral is

8

360°. 15.

A square is a rectangle.

16.

The diagonals of an isosceles trapezoid have the same length.

17.

In parallelogram ABCD, angles A and B are supplementary.

18.

In parallelogram ABCD, angles A and B are congruent.

19.

The sum of the exterior angles, one at each vertex, is the same for an octagon as it is for a triangle.

20.

A hexagon is also an octagon.

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Geometry Practice

Chapter 6: Polygons and Quadrilaterals

Name:

Date:

Chapter 6: Polygons and Quadrilaterals (cont.) Directions: For questions 21–30, state whether the following statements are true [T] or false [F]. 21.

A rhombus is a square.

22.

The sum of the exterior angles of a pentagon, one at each vertex, is 360°.

23.

The diagonals of a square bisect the angles of the square.

24.

Opposite angles of a parallelogram are congruent.

25.

In parallelogram ABCD, if the measure of angle A is equal to the measure of angle B, then ABCD must be a square.

26.

The measure of each exterior angle in a regular octagon is 45°.

27.

A trapezoid has two angles that are equal.

28.

If the diagonals of a rectangle are perpendicular, then the rectangle is a square.

29.

If the diagonals of a parallelogram are perpendicular, then the parallelogram is a square.

30.

The diagonal for a parallelogram is a line of symmetry for the parallelogram.

31.

In quadrilateral ABCD, if the measures of angles A, B, and C are 42°, 55°, and 102°, respectively, what is the measure of angle D?

32.

Soma measured the angles in a regular pentagon and found them to be 72°. Is he correct?

33.

If the measure of each angle of a regular polygon is 156°, how many sides does the polygon have?

34.

If a regular polygon has 11 sides, what is the measure of each angle?

35.

What is the sum of the measures of the angles of a regular dodecagon [12 sides]?

36.

In parallelogram ABCD, the measure of angle A is 74°. What are the measures of the other angles?

37.

In a rectangle that is 9 cm by 11 cm, how long is the diagonal?

38.

T or F? A trapezoid can have three sides of equal length.

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Geometry Practice

Chapter 6: Polygons and Quadrilaterals

Name:

Date:

Chapter 6: Polygons and Quadrilaterals (cont.) Challenge Problems: Polygons and Quadrilaterals Directions: State whether the following statements are always [A], sometimes [S], or never [N] true. 1.

The sum of the measures of the exterior angles, one at each vertex, of a hexagon is 360°.

2.

An equilateral quadrilateral is a square.

3.

The diagonals of a trapezoid bisect each other.

4.

The diagonals of a parallelogram bisect the angles of the parallelogram.

5.

The diagonals of a rhombus are perpendicular.

6.

The measure of each exterior angle in a regular hexagon is 60°.

Directions: Circle the correct answer for the following True/False questions. 7.

T or F? If the diagonals of rhombus are equal in length, the rhombus is a square.

8.

T or F? If the diagonals of a parallelogram are equal in length, the parallelogram is a rectangle.

9.

T or F? A square is a rhombus.

10.

T or F? The diagonal of a square is a line of symmetry for the square.

11.

T or F? A diagonal of a parallelogram divides the parallelogram into two congruent triangles.

12.

T or F? When both diagonals of a parallelogram are drawn, they divide the parallelogram into four congruent triangles.

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Geometry Practice

Chapter 6: Polygons and Quadrilaterals

Name:

Date:

Chapter 6: Polygons and Quadrilaterals (cont.) 13.

Complete the following table regarding properties of quadrilaterals. Place a YES in a cell to indicate the property holds for the quadrilateral listed and place a NO if it does not hold. Quadrilateral

Opposite angles are equal





Consecutive angles are supplementary





Opposite sides are equal







Diagonals bisect each other









Diagonals are perpendicular to each other





Diagonals bisect the angles









Each angle measures 90°









Diagonals are equal











Has at least one line of symmetry



Is o Tr sce ap le ez s oi d

ar

e

le Sq u

R

ec

ta

ng

bu s m

R







ho

Property

Pa r



al

le

lo

gr

am











14.

What is the sum of the measures of the angles of a regular 22-gon?

15.

What is the sum of the measures of the angles of a regular n-gon?

16.

T or F? If the midpoints of the sides of a quadrilateral are joined in order, the resulting figure is a parallelogram.

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74

Geometry Practice

Chapter 6: Polygons and Quadrilaterals

Name:

Date:

Chapter 6: Polygons and Quadrilaterals (cont.) Checking Progress: Polygons and Quadrilaterals 1.

State whether the following statement is always [A], sometimes [S], or never [N] true:



If the diagonals of a quadrilateral bisect each other, the quadrilateral is a parallelogram.

2.

State whether the following statement is always [A], sometimes [S], or never [N] true:



A parallelogram has a line of symmetry.

3.

State whether the following statement is always [A], sometimes [S], or never [N] true:



A square is a rectangle.

4.

T or F? A trapezoid may have a right angle.

5.

T or F? A rhombus is a regular quadrilateral.

6.

In a rectangle that is 9 cm by 12 cm, how long is the diagonal?

7.

In parallelogram ABCD, the measure of angle A is 44°. What are the measures of the other angles?

8.

In quadrilateral ABCD, if the measures of angles A, B, and C are 52°, 65°, and 92°, respectively, what is the measure of angle D?

9.

What is the measure of each angle in a regular decagon [10 sides]?

10.

What is the sum of the measures of the angles in a 15-gon?

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Chapter 7: Circles

Geometry Practice

Name:

Date:

Chapter 7: Circles Basic Overview: Radius, Diameter, Circumference and Area, Chords, Tangents, Secants, Arcs, Inscribed Angles, Finding the Equation of a Circle A circle is defined as the set of all points in a plane that are equidistant from a given point, called the center of the circle. Points related to a circle can be inside the circle (interior points), outside the circle (exterior points), and points on the circle itself. The radius of a circle is a line segment with one endpoint at the center of the circle and the other endpoint on the circle. The diameter of a circle is a line segment that passes through the center point and has both endpoints on the circle. All diameters in a single circle are equal. The circumference is the distance around the circle. To calculate the circumference (C) find the diameter (d) and multiply the diameter by pi (π), that is C = πd [or in terms of the radius, C = 2πr ]. Two theorems related to circles are: (1) two circles are congruent if and only if their diameters are the same length; and (2) two circles are congruent if and only if their radii are the same length. The area of a circle (A) is found by multiplying pi (π) times the square of the radius d 2 (r 2) A = πr 2 or pi times !s the diameter squared, A = π . If you know the circumference, the 2 d 2 area can be found by first finding the diameter of the circle and then using A = π . To find 2 the diameter if you know the circumference, divide the circumference by pi because C = πd. Two circles are congruent if: the radii are the same length, diameters are the same length, circumferences are the same, or the areas are equal. A chord of a circle is a line segment with endpoints on the circle. Chords are different than the radius and diameter because in a single circle, not all chords are always the same length. A chord does not have to pass through the center of the circle. Note that the diameter is also a special chord. The diameter of a circle is the longest chord in the circle. A tangent is a line that touches the circle in exactly one point. The point of intersection is called the point of tangency. A secant line for a circle is a line that intersects the circle in two points or contains a chord of the circle. Theorems related to tangents include: (1) if a line is tangent to a circle, the tangent is perpendicular to the radius drawn to the point of tangency; and (2) conversely, if a line is perpendicular to a radius of a circle at its endpoint on the circle, then the line is tangent to the circle. Arcs can be measured in degrees by measuring the central angle that is made when two radii intersect with the two endpoints of the arc. If the arc is a minor arc (less than 180°), then the degree-measure of the arc is equal to the measure of the arc. If the arc is a major arc (greater than 180°), then subtract the degree-measure of the minor arc from 360° to get the degree-measure of the major arc. The length of the arc can also be determined by using a proportional length of the circumference of the circle.

( )

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76

( )

Chapter 7: Circles

Geometry Practice

Name:

Date:

Chapter 7: Circles (cont.) An angle formed by two chords that have a common endpoint is called an inscribed angle. The arc that lies in the interior of the inscribed angle is called the intercepted arc. The measure of an inscribed angle is half the degree-measure of its intercepted arc. Using a coordinate plane, you can find an algebraic equation for a circle. The standard equation of a circle with the radius r and the center (h, k) is (x – h)2 + (y – k)2 = r 2. Examples of Radius, Diameter, Circumference and Area, Chords, Tangents, Secants, Arcs, Inscribed Angles, Finding the Equation of a Circle

Examples of the Regions of a Circle:







Examples of Circle, Radius, and Diameter:





AB, AC, AD, and AE are radii, and they are equal.





BD is a diameter of circle A.





Notice that in the circle at the right, the diameter is twice as long as the radius. A formula for finding the diameter is d = 2r.

A is an interior point. B is a point on the circle. C is an exterior point.

Examples of Finding the Circumference: Find the circumference of a circle with the diameter of 4 cm. C = πd







C = πd

C ≈ 3.14(4)





C ≈ SuS(4)

C ≈ 12.56





C ≈ KuK





Find the circumference of a circle with the radius of 2 cm. C = π2r







C = π2r

C ≈ 3.14(2)(2)





C ≈ SuS (2)(2)

C ≈ 3.14(4)





C ≈ SuS (4)

C ≈ 12.56





C ≈ KuK



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77

Chapter 7: Circles

Geometry Practice

Name:

Date:

Chapter 7: Circles (cont.) Examples of Finding the Area of a Circle:

Find the area of circle P.



If you know the radius PM = 3 cm:

If you know the diameter LN = 6 cm:

A = πr 2



( d2 )





A=π

A ≈ 3.14(32)



A ≈ 3.14

A ≈ 3.14(9)



A ≈ 3.14(9)

A ≈ 28.26



A ≈ 28.26



2

( 62 )

If you know the circumference ≈ 18.85: Find the diameter. 18.85 ≈ 3.14d 18.85 ≈ 3.14(d) 3.14 3.14 6≈d Find the area. d 2 A=π 2

( )

( 62 )

A ≈ 3.14

2

A ≈ 3.14(9) A ≈ 28.26

© Mark Twain Media, Inc., Publishers

78

2

Chapter 7: Circles

Geometry Practice

Name:

Date:

Chapter 7: Circles (cont.) Example of Finding the Length of the Arc: Measure of the Arc CD is 30° A

Diameter AC = 6. B

Find the circumference. C = πd

C

C = (π)6 Divide the measure of the arc by 360°. 30 = 1 360 12

aQs).

Multiply the circumference (6π) times the quotient (

(aQs).

The length of Arc CD = 6π The length of Arc CD =

≈ 3.14 = 1.57. !s π 2

Example of an Inscribed Angle:

Arc AC is the intercepted arc. Chords AB and BC form the inscribed angle.

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79

D

Geometry Practice

Chapter 7: Circles

Name:

Date:

Chapter 7: Circles (cont.) Example of Finding the Measure of an Inscribed Angle: Find the measure of the inscribed angle ABC. Measure the central angle. ∠ADC = 75°, which means the Arc AC is a minor arc.

D

Therefore, the degree-measure of Arc AC is also 75°. Divide the degree-measure of the Arc AC by 2. 75 = 37.5 2 The measure of the inscribed angle is 37.5°. Examples of a Circle in a Coordinate Plane:



Use a coordinate plane to find the algebraic equation for this circle.

Center is (10, 10)

Radius = 7

Substitute the coordinates for the center of the circle into the formula. (x – h)2 + (y – k )2 = r 2 (x – 10)2 + (y – 10)2 = 72 Simplified: (x – 10)2 + (y – 10)2 = 49

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80

Geometry Practice

Chapter 7: Circles

Name:

Date:

Chapter 7: Circles (cont.) Practice: Radius, Diameter, Circumference and Area, Chords, Tangents, Secants, Arcs, Inscribed Angles, Finding the Equation of a Circle Directions: State whether the following statements are always [A], sometimes [S], or never [N] true. 1.

A diameter is the longest chord in a circle.

2.

A radius is a chord of a circle.

3.

If two chords intersect inside a circle, they bisect each other.

4.

Two different diameters of a circle are perpendicular to each other.

5.

If a tangent and a diameter intersect, they intersect at right angles.

Directions: Draw the following circles. 6.

Draw a circle with two perpendicular radii.



7.

Draw a circle with two parallel secants.

8.

Draw a circle with a secant perpendicular to a diameter but not at the center of the circle.

9.

Draw a circle with a tangent perpendicular to a diameter.

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81

Geometry Practice

Chapter 7: Circles

Name:

Date:

Chapter 7: Circles (cont.) 10.

Draw a circle with a tangent parallel to a diameter.



11.

12.

Draw a circle with two radii that are parallel.

Draw a circle with a secant bisecting a radius.

Directions: Complete problems 13–17 on the coordinate grids provided. 13.

a.

Sketch a circle with center (2, 9) and radius 5.



b.

What are the endpoints of the diameter that is parallel to the y-axis?

c.

List five other points on this circle with integer coefficients.



d.

What is the equation for the circle? y

14 12 10 8 6 4 2 0 -4 -2 0 2 4 6 8 10 12 14 16 18 20 22 24 -2

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82

x

Chapter 7: Circles

Geometry Practice

Name:

Date:

Chapter 7: Circles (cont.) 14.

a.

Sketch a circle with the endpoints for its diameter at (11, 9) and (23, 9).

b.

What is the length of the radius?

c.

What is the equation for the circle? y

16 14 12 10 8 6 4 2 0 -4 -2 0 2 4 6 8 10 12 14 16 18 20 22 24 -2

y

15.

a.

Sketch a circle





with the endpoints





for its diameter at

18





(5, 9) and (11, 17).

16

b.

What is the





length of the





radius?

c.

What is the





equation for the





circle?





© Mark Twain Media, Inc., Publishers

x

20

14 12 10 8 6 4 2 0 -4 -2 0 2 4 6 8 10 12 14 16 18 20 22 24 -2

83

x

Chapter 7: Circles

Geometry Practice

Name:

Date:

Chapter 7: Circles (cont.) y

16.

a.

Sketch a circle





with a diameter of





10, and one point





on the circle is (0, 12).

12

b.

What is the area

10





of the circle?





16 14

8 6 4 2 0 -4 -2 0 2 4 6 8 10 12 14 16 18 20 22 24 -2

17.

a.

Draw a circle that is tangent to the x-axis at point (4, 0) and y-axis at point (0, 4).



b.

What is the area of the circle?

c.

What is the circumference of the circle?

d.

How far is the center of the circle from (0, 0)? y 10 8 6 4 2 0 -4 -2 0 2 4 6 8 10 12 14 16 18 20 22 24 -2

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84

x

x

Chapter 7: Circles

Geometry Practice

Name:

Date:

Chapter 7: Circles (cont.) Directions: Draw the following circles in the space provided. 18.

a.

Draw a circle with a central angle of 70°.

b.

What is the degree-measure of the intercepted arc?

c.

If the circle had a radius of 5, what is the length of the intercepted arc?

19.

Draw a circle with a central angle of 50°.

a.

b.

What is the degree-measure of the intercepted arc?

c.

If the circle had a radius of 5, what is the length of the intercepted arc?

20.

Draw a circle with an inscribed angle of 70°.

a.

b.

What is the degree-measure of the intercepted arc?

c.

If the circle had a radius of 5, what is the length of the intercepted arc?

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85

Chapter 7: Circles

Geometry Practice

Name:

Date:

Chapter 7: Circles (cont.) 21.

For a circle with a radius of 12 cm, what is the



a.

22.

For a circle with a diameter of 5 cm, what is the



a.

23.

For a circle with a circumference of 20 cm, what is the radius?

24.

What is the center and radius of the circle with the following equation?



(x + 2)2 + (y – 1)2 = 5

circumference?

circumference?





b.

b.

area?

area?



25.

a.

If a central angle cuts off an arc of 90°, what is the measure of the inscribed angle?



b.

On the same circle, draw four inscribed angles that intercept the same 90° arc.

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86

Chapter 7: Circles

Geometry Practice

Name:

Date:

Chapter 7: Circles (cont.) Challenge Problems: Circles 1.

Draw a circle with a central angle of 60°.

2.

What is the degree-measure of the intercepted arc in question 1?

3.

In question 1, if the circle had a radius of 10 cm, what is the length of the intercepted arc?

4.

For a circle with a radius of 4.5 cm, what is the circumference?

5.

For a circle with a radius of 4.5 cm, what is the area?

6.

For a circle with a diameter of 1.2 cm, what is the circumference?

7.

For a circle with a diameter of 1.2 cm, what is the area?

8.

What is the center and radius of the circle with the following equation?



(x + 5)2 + (y – 6)2 = 51

9.

What is the equation of the circle with radius 7 cm and center at the origin?

10.

For a circle with area of 225 cm2, what is the length of the radius?

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87

Geometry Practice

Check-Up Problems: Coordinate Geometry

Name:

Date:

Check-Up Problems: Coordinate Geometry Directions: Work these problems on your own paper and write the answers below. Use your own graph paper to graph the lines. 1.

Find the midpoint of BC where B = (5, 3) and C = (1, -2)

2.

Find the midpoint of AD where A = (4, -2) and D = (5, 12)

For problems 3–4, assume D is the midpoint of AE. 3.

AD has a length of 7. What is the length of DE?

4.

DE has a length of 12. What is the length of AE?

5.

Find the slope of FG where F = (7, 4) and G = (-2, 7).

6.

Find the slope of HI where H = (-7, -4) and I = (2, -7).

7.

Are the lines in questions 5 (FG) and 6 (HI) parallel, perpendicular, or neither? Explain.

8.

Find the slope of the line and graph. 3y – 6x = 18

9.

Find the equation of the line and graph. Slope is 5 and one point is (-3, 0).

10.

Find the equation of the line and graph. Points (3, 5) and (6, 7)

11.

Find the distance between points A and C between (1, 4) and (5, 16).



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88

Geometry Practice

Check-Up Problems: Measuring and Classifying Angles

Name:

Date:

Check-Up Problems: Measuring and Classifying Angles Directions: Draw the following angles in the space provided. Write the measure of the angle on each drawing. 1.

Draw an example of a right angle. What is the measure of the angle?

2.

Draw an example of an obtuse angle. What is the measure of the angle?

3.

Draw an example of an acute angle. What is the measure of the angle?

4.

Draw an example of a reflex angle. What is the measure of the angle?

5.

Draw an angle and show 4 different ways to label the angle.

6.

Draw an example of supplementary angles. What are the measures of the angles?

8.

Draw an example of vertical angles. 9. What are the measures of the angles?

Draw an example of an angle bisector. What are the measures of the angles?

10.

Draw an example of a straight angle. What is the measure of the angle?

Draw an example of congruent angles. What are the measures of the angles?

12.

Draw an example of adjacent angles. What are the measures of the angles?

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7.

11.

89

Draw an example of complementary angles. What are the measures of the angles?

Geometry Practice

Check-Up Problems: Triangles

Name:

Date:

Check-Up Problems: Triangles Directions: Answer the following questions. Use your own paper if you need more room to work a problem. 1.

State whether the following statement is always [A], sometimes [S], or never [N] true:



In an obtuse triangle, the longest side is opposite the obtuse angle.

2.

State whether the following statement is always [A], sometimes [S], or never [N] true:



A scalene triangle is isosceles.

3.

T or F? If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the triangles are congruent.

4.

T or F? The sum of the exterior angles of a triangle, one at each vertex, is the same as the sum of the interior angles of a square.

5.

If one leg of a right triangle is 12 cm and the hypotenuse is 17 cm, what is the length of the other leg?

6.

If one leg of a right triangle is 12 cm and the hypotenuse is 17 cm, what is the area of the triangle?

7.

Find the point P, which is the midpoint between (2, 15) and (14, 31).



Now find the midpoint between (2, 5) and P.

8.

If (14, 2b) is the midpoint between (4a, 7) and (12, 21), what are the values for a and b?

9.

If 5 cm, 18 cm, and x are the lengths of the sides of a triangle. What is the smallest value possible for the perimeter?

10.

What is the largest value of the perimeter?

Draw two triangles with a perimeter of 18 cm and different areas. What are the areas of each triangle?

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90

Geometry Practice

Check-Up Problems: Polygons and Quadrilaterals

Name:

Date:

Check-Up Problems: Polygons and Quadrilaterals Directions: Answer the following questions. Use your own paper if you need more room to work problems. 1.

State whether the following statement is always [A], sometimes [S], or never [N] true:



An isosceles trapezoid has two pairs of congruent angles.

2.

State whether the following statement is always [A], sometimes [S], or never [N] true: When the midpoints of a rectangle are connected in order, the resulting quadrilateral is a rhombus.

3.

State whether the following statement is always [A], sometimes [S], or never [N] true: When the midpoints of a rhombus are connected in order, the resulting quadrilateral is a square.

4.

T or F? If the diagonals of a parallelogram are equal in length, the parallelogram is a rectangle.

5.

T or F? A square is a regular quadrilateral.

6.

In a rectangle that is 18 cm by 24 cm, how long is the diagonal?

7.

In parallelogram ABCD, the measure of angle A is 64°. What are the measures of the other angles?

8.

In quadrilateral ABCD, if the measures of angles A, B, and C are 47°, 70°, and 82°, respectively, what is the measure of angle D?

9.

What is the measure of each angle in a regular 18-gon?

10.

What is the sum of the measures of the angles in a 25-gon?

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91

Check-Up Problems: Circles

Geometry Practice

Name:

Date:

Check-Up Problems: Circles Directions: Answer the following questions. Use your own paper if you need more room to work problems. 1.

Draw a circle with an inscribed angle of 60°.

2.

What is the degree-measure of the intercepted arc in question 1?

3.

In question 1, if the circle had a radius of 10, what is the length of the intercepted arc?



4.

For a circle with a radius of 1.1 cm, what is the circumference?

5.

For a circle with a radius of 1.1 cm, what is the area?

6.

For a circle with a diameter of 44 cm, what is the circumference?

7.

For a circle with a diameter of 44 cm, what is the area?

8.

What is the center and radius of the circle with the following equation?



(x – 2)2 + (y + 6)2 = 51

9.

What is the equation of the circle with a diameter of 7 cm and the center at the origin?



10.

For a circle with circumference of 288 cm, what is the length of the radius?



© Mark Twain Media, Inc., Publishers

92

Practice Answer Keys

Geometry Practice

Practice Answer Keys Chapter 1: Practice: Shapes, Congruence, Similarity, and Symmetry (pages 16–17) 1. Octagon Drawing: Drawings will vary, but the polygon must have eight sides.

Explanation: The polygon has eight sides.

2.

Hexagon



Explanation: The polygon has six sides.

3.

Triangle



Explanation: The polygon has three sides.

4.

Quadrilateral Drawing: Drawings will vary, but the polygon must have four sides.



Explanation: The polygon has four sides.

5.

A and C are congruent; B and D are similar; A and C are also similar.

Drawing: Drawings will vary, but the polygon must have six sides.

Drawing: Drawings will vary, but the polygon must have three sides.

Possible solutions include: 6.





7.

8.





9.











© Mark Twain Media, Inc., Publishers

93

Practice Answer Keys

Geometry Practice

Practice Answer Keys (cont.) Chapter 1: Challenge Problems: Shapes, Congruence, Similarity, and Symmetry (page 18) 1.

Answers will vary.

2.

Answers will vary.

3.

Answers will vary.

4.

Answers will vary.

Chapter 1: Checking Progress: Shapes, Congruence, Similarity, and Symmetry (pages 19–20) 1.

Any one-dimensional polygon

2.

Any three-dimensional polyhedron

3.

Answers will vary.

4.

Answers will vary.

5.

Answers will vary.

Drawings will vary. Examples are given. Polygon Drawing

Number of Sides

6.

Hexagon











6

7.

Square











4

8. 9.

Circle Quadrilateral









0 4

10.

Octagon









8



© Mark Twain Media, Inc., Publishers

94

As a side is a line segment, a circle is not considered to have a side.

Practice Answer Keys

Geometry Practice

Practice Answer Keys (cont.) 11.

Pentagon











5

12.

Triangle











3

13.

Rectangle









4

Chapter 2: Practice: Midpoint Formulas, Slope, Equations, Distance (pages 27–30) 1.

Midpoint of AB is (2, 4)

2.

Midpoint of CD is (1, 1)

3.

Midpoint of EF is (3, 4)

4.

Midpoint of GH is (3, 6)

5.

m=-

!d









6.

© Mark Twain Media, Inc., Publishers

#d = -1

11 10 9 8 7 6 5 4 3 2 1 0 -11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 -11

11 10 9 8 7 6 5 4 3 2 1 0 -11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 -11

B

m=-

D

A

C

95

Geometry Practice

Practice Answer Keys

Practice Answer Keys (cont.) 7.

m=2

11 10 9 8 7 6 5 4 3 2 1 0 -11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 -11

F

E

!d

@d

8.

Slope of AB = ; Slope of CD = -3

9.

Slope of EF = 3; Slope of GH = -



Perpendicular



Neither parallel nor perpendicular







11 10 9 8 7 6 5 4 3 2 1 0 -11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 -11

11 10 9 8 7 6 5 4 3 2 1 0 -11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 -11

A

C

G

H

B

F

D

© Mark Twain Media, Inc., Publishers

E

96

Geometry Practice

Practice Answer Keys

Practice Answer Keys (cont.) 10.

!d

!d

Slope of IJ = ; Slope of KL =

Parallel

11 10 9 8 7 6 5 4 3 2 1 0 -11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 -11

L

K

J

I

11.

3x – y + 2 = 0 y = 3x + 2 y-intercept = (0, 2) Slope = 3







12.

Slope = -

%s

11 10 9 8 7 6 5 4 3 2 1 0 -11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 -11

11 10 9 8 7 6 5 4 3 2 1 0 -11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 -11

(1, 5)

(0, 2)

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5x + 2y = -8 2y = -5x – 8 y = -5x – 8 2 y-intercept (0, -4)

(0, -4)

(2, -9)

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Geometry Practice

Practice Answer Keys

Practice Answer Keys (cont.) 13.

y + 4 – 4x = 0 y = 4x – 4



y-intercept = (0, -4) Slope = 4

11 10 9 8 7 6 5 4 3 2 1 0 -11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 -11

(1, 0)

(0, -4)

14.

Slope = 3, Point (4, -4) y = mx + b -4 = 3x + b b = -3(4) – 4 b = -16



y = 3x – 16

15.

Slope = , Point (0, -1) y = mx + b



-1 = (0) + b b = -1



y=

16.

(2, 3) and (6, 2) y=3–2=1 x = 2 – 6 = -4 Slope = -



y=- x+b

!f

!f



!f x – 1 !f

!f 3 = - !f (2) + b 3!s = b y = - !fx + 3 !s

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Geometry Practice

Practice Answer Keys

Practice Answer Keys (cont.) 17.

(1, 6) and (2, 1) y=6–1=5 x = 1 – 2 = -1



Slope = -



y = mx + b y = -5x + b 6 = -5(1) + b 11 = b



y = -5x + 11

%a = -5

18 (6, -1) and (4,-2) y = -1 – (-2) = 1 x=6–4=2

!s



Slope =



y = mx + b



-1 = (6) + b b = -4



y= x–4

19.

(3, 2) and (1, 8)



(x2 – x1)2 + (y2 – y1)2



(1 – 3)2 + (8 – 2)2



(-2)2 + (6)2



4 + 36



40 = 2 10

!s

!s

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Practice Answer Keys

Geometry Practice

Practice Answer Keys (cont.) 20.

(-3, 2) and (4, -1)



(x2 – x1)2 + (y2 – y1)2



[4 – (-3)]2 + (-1 – 2)2



(7)2 – (3)2



49 + 9



58

Chapter 2: Challenge Problems: Midpoint Formulas, Slope, Equations, Distance (page 31) 1.

y – 3x = 1 y + 3x = -2 D. Neither

2.

y = 3x – 3

3.

y+4= x

4.

!d

y = - x + 2 C. Perpendicular

!d y – 1 = !dx B. Parallel y = -!dx – 2 y = -!dx + 2 A. Parallel

Chapter 2: Checking Progress: Coordinate Geometry (page 32)

(

) (

(

) (

)

1.

Midpoint of BC = 6 + 1 , 3 + -2 = 7 , 1 2 2 2 2

2.

Midpoint of AD = 3 + 5 , -2 + 12 = 8 , 10 2 2 2 2

3.

Length of DE: 10 units

4.

Length of AE: 22 units

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) = (4, 5)

Practice Answer Keys

Geometry Practice

Practice Answer Keys (cont.)

!s

5.

m=-

6.

m=2

7.

FG is perpendicular to HI. The slopes are negative reciprocals, and m1 • m2 = -1.

8.

4y – 8x = 16 4y = 8x + 16 y = 2x + 4 Slope is 2, y-intercept is (0, 4) 11 10 9 8 7 6 5 4 3 2 1 0 -11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 -11

(1, 6) (0, 4)

9.

y = mx + b 0 = -7(-3) + b -21 = b y = -7x – 21 22 20 18 16 14 12 10 8 6 4 2 0 -22 -20-18-16-14-12-10-8 -6 -4 -2 0 2 4 6 8 10 12 14 16 18 20 22 -2 -4 -6 -8 -10 -12 -14 -16 -18 -20 -22

(-3, 0)

(0, -21)

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Practice Answer Keys

Geometry Practice

Practice Answer Keys (cont.) 10.

y = 5 – 7 = -2 x = 4 – 6 = -2



y = mx + b y = 1x + b 5 = 1(4) + b b=1 y = 1x + 1

Slope = -2 = 1 -2

11 10 9 8 7 6 5 4 3 2 1 0 -11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 -11

(6, 7)

(4, 5)

11.

Distance between points A (7, 7) and C (1, 4)



(x2 – x1)2 + (y2 – y1)2



(7 – 1)2 + (7 – 4)2



(6)2 + (3)2



36 + 9



45 = 3 5

Chapter 3: Practice: Measuring and Classifying Angles (pages 38–39) 1.

m∠ = 100°

2.

m∠ = 35°

3.

m∠ = 120°

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Practice Answer Keys

Geometry Practice

Practice Answer Keys (cont.) 4.

m∠ = 300°

5.

m∠ = 10°

6.

Drawings will vary. Angle is less than 90 degrees.

7.

Drawings will vary. Angle is greater than 90 degrees and less than 180 degrees.

8.

Drawings will vary. Angle is 90 degrees.

9.

Drawing is a straight line with points labeled. Angle is 180 degrees.

10.

Drawings will vary. Angle is greater than 180 degrees and less than 360 degrees.

11.

Drawings will vary. The two angles add up to 90 degrees.

12.

Drawings will vary. The two angles share a common side and do not have common interior points.

13.

Drawings will vary. The two angles are adjacent, and non-common sides are opposite rays.

14.

Drawings will vary. The two angles have the same measure.

15.

Drawings will vary. The angles’ sides form two pairs of opposite rays.

Chapter 3: Challenge Problems: Measuring and Classifying Angles (pages 40–42) 1.

There are three possible angles, namely: ∠EGH, ∠EGF, and ∠FGH, that include the point, G, as the vertex. It is best to use three letters to differentiate among the three possibilities.

2.

Interior – point C; Exterior – point A; on the angle – points B and D

3.

You can label the angle as: ∠F, ∠GFH, ∠HFG, or ∠3.

4.

A.

Vertical Angles: Angles 1 and 3 are equal, and 2 and 4 are equal.



B.

Linear Pairs: ∠ADB and ∠BDC form a linear pair because they are supplementary. They are also adjacent angles because they share a common side.

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Geometry Practice

Practice Answer Keys

Practice Answer Keys (cont.)

C.

Adjacent Angles: ∠ABD is adjacent to ∠DBC because they share the common side BD and do not have any common interior points.



D.





Angle Bisector: The angle bisector is BD because it divides ∠ABC into two equal parts. Also Adjacent Angles



E.

Congruent Angles: ∠ABD and ∠EBC both have the same measure, so they are congruent. ∠ABE and ∠DBC both have the same measure, so they are also congruent.



F.





Supplementary Angles: ∠ABD and ∠DBC are supplementary because they form a straight line, which adds up to 180 degrees. Also Adjacent Angles



G.

Complementary Angles: ∠ABD and ∠DBC add up to 90 degrees. Also Adjacent Angles

Chapter 3: Checking Progress: Measuring and Classifying Angles (page 43) 1.

H. right angle



2.

J. supplementary angles

3.

D. congruent angles

4.

A. acute angle

5.

C. complementary angles

6.

K. vertical angles

7.

F. obtuse angle

8.

G. reflex angle

9.

B. angle bisector

10.

I. straight angle

11.

E. linear pair

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Geometry Practice

Practice Answer Keys

Practice Answer Keys (cont.) Chapter 4: Practice: Patterns, Structure of Geometry, Conditional Statements, and Deductive Reasoning (pages 46–47) 1.

#of Rays (n) # of non-reflex Angles

2 1

3 3

4 6

5 10

6 15



The number of angles that 12 rays with a common endpoint make is 66: Each ray adds n – 1 to the previous number of angles.

2.

3.

0, 4, 8, 12, 16

4.

Hypothesis: if someone pays $25 Conclusion: I will sell the table Converse: If I will sell the table, then someone pays $25.

5.

Hypothesis: if you are willing Conclusion: you are able Converse: If you are able, then you are willing.

6.

Hypothesis: if two angles have the same measurement Conclusion: they are congruent Converse: If two angles are congruent, then they have the same measurement.

7.

Two angles are adjacent if and only if they share a common side and have no common interior points.

8.

A ray is a bisector ray if and only if it divides the angle into two equal parts.

9.

True; For example, the point where two lines meet is on both lines.

10.

False; x can be +2 or -2.

11.

x 2 = 9 if and only if x is +3 or -3.

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Geometry Practice

Practice Answer Keys

Practice Answer Keys (cont.) Chapter 4: Challenge Problems: Patterns, Structure of Geometry, Conditional Statements, and Deductive Reasoning (pages 48–49) 1.

a∠1 (n )

5°

15°

25°

35°

45°

55°

65°

75°

85°



a∠2

85°

75°

65°

55°

45°

35°

25°

15°

5°



90 – n = a∠2



y 85 80 75 70 65 60 55 50 45 40 35 30 25 20 15 10 5 0 -10 -5 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 -5

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x

Practice Answer Keys

Geometry Practice

Practice Answer Keys (cont.) 2.

∠ABD and ∠DBC are linear pairs; the sum of linear pairs = 180° m∠ABC = 180° m∠ABC = m∠ABD + m∠DBC ∠ABD = 140° m∠ABC = 180° = 140° + m∠DBC 180° – 140° = m∠DBC 40° = m∠DBC

3.

m∠AOB + m∠BOC = m∠AOC 60° + m∠BOC = 100° m∠BOC = 100° – 60° m∠BOC = 40°

Chapter 4: Checking Progress: Patterns, Structure of Geometry, Conditional Statements, and Deductive Reasoning (page 50) 1.

Unit, repetition, and a system of organization

2.

Answers will vary.

3.

Postulates

4.

theorem

5.

conditional

6.

Bi-conditional

7.

Deductive

8.

Inductive

9.

conjectures

10.

logical

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Practice Answer Keys

Geometry Practice

Practice Answer Keys (cont.) Chapter 5: Practice: Triangles (pages 57–59) 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

A S S N S N S A A S A N S S A

16. 17. 18. 19. 20.

T T F T T

21.

13 cm

22.

119 cm

23.

5 2 cm

24.

%s 2 cm

25.

5 3 cm

26.

(10, 8)

27.

a = 4, b = 3

28.

a = -7, b = 14

29.

a = 11, b = -19

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Practice Answer Keys

Geometry Practice

Practice Answer Keys (cont.) 30.

a.

120 square units



b.

3 units



c.

20 units



d.

30 square units



e.

409 units

31.

3c

32.

Perimeter is 24 cm; area is 24 square centimeters.

33.

Each triangle will have an area of 20 square units.

34.

Perimeter is 60 cm; area is 150 square centimeters.

35.

24 + 12 2 cm

Chapter 5: Challenge Problems: Triangles (pages 60–61) 1.

A

2.

This can be drawn and answers will vary.

3.

This is not possible.

4.

This is not possible.

5.

This can be drawn and answers will vary, but the triangle must be isosceles.

6.

F

7.

T

8.

F [If the angle is included between the two sides, the statement is true]

9.

T

10.

T

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Practice Answer Keys

Geometry Practice

Practice Answer Keys (cont.) 11.

T

12.

10 154 cm

13.

1, 4, 4; 2, 3, 4; and 3, 3, 3

14.

17 cm; 27 cm

15.

15 cm

Chapter 5: Checking Progress: Triangles (page 62) 1.

S

2.

S

3.

F [This is true if the parts correspond, but false if they do not.]

4.

T

5.

193 cm

6.

42 cm2

7.

2 + a , 5 + b ( 2 2 )

8.

a = 13 and b = 19

9.

31 cm; 49 cm

10.

Answers will vary.

Chapter 6: Practice: Polygons and Quadrilaterals (pages 71–72) 1. 2. 3. 4. 5. 6. 7.

A S S [A square rectangle has 4.] S S [If it is a square.] A S

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Practice Answer Keys

Geometry Practice

Practice Answer Keys (cont.) 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.

S A S S S S A A A A S A N

21. 22. 23. 24. 25. 26. 27. 28. 29. 30.

F T T T F T F T F F

31.

161°

32.

No, they should measure 108°.

33.

15 sides

34.

180 – 360 )° or about 147.27° ( 11

35.

1,800°

36.

The measures of angles B, C, and D are 106°, 74°, and 106°, respectively.

37.

202 cm

38.

T

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Geometry Practice

Practice Answer Keys

Practice Answer Keys (cont.) Chapter 6: Challenge Problems: Polygons and Quadrilaterals (pages 73–74) 1. 2. 3. 4. 5. 6.

A S N S A N

7. 8. 9. 10. 11. 12.

T T T T T F Quadrilateral

Opposite angles are equal.



re ua

Is o Tr sce ap le ez s oi d

le Sq

R

ec

ta

ng

bu s m

ho R

Property

Pa r



al

le

lo

gr

am

13.



YES YES YES YES NO

Consecutive angles are supplementary.

YES YES YES YES NO

Opposite sides are equal.





YES YES YES YES NO

Diagonals bisect each other.





YES YES YES YES NO

Diagonals are perpendicular to each other.

NO YES NO YES NO

Diagonals bisect the angles.





NO YES NO YES NO

Each angle measures 90°.





NO NO YES YES NO

Diagonals are equal.





NO NO YES YES YES



NO YES YES YES YES



Has at least one line of symmetry 14. 15. 16.

3,600° [(n – 2)180]° T

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Geometry Practice

Practice Answer Keys

Practice Answer Keys (cont.) Chapter 6: Checking Progress: Polygons and Quadrilaterals (page 75) 1.

A

2.

S

3.

A

4.

T

5.

F

6.

15 cm

7.

The measures of angles B, C, and D are 136°, 44°, and 136°, respectively.

8.

151°

9.

144°

10.

2,340°

Chapter 7: Practice: Radius, Diameter, Circumference and Area, Chords, Tangents, Secants, Arcs, Inscribed Angles, Finding the Equation of a Circle (pages 81–86) 1.

A

2.

N

3.

S

4.

S

5.

A

6.

Answers will vary. An example is:

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Geometry Practice

Practice Answer Keys

Practice Answer Keys (cont.) 7.

Answers will vary. An example is:

8.

Answers will vary. An example is:

9.

Answers will vary. An example is:

10.

Answers will vary. An example is:

11.

Answers will vary. An example is:

12.

This cannot be done as all radii have one endpoint at the center of the circle.

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Geometry Practice

Practice Answer Keys

Practice Answer Keys (cont.) 13.

y

a.

14 12 10 8 6 4 2 0 -4 -2 0 2 4 6 8 10 12 14 16 18 20 22 24 -2



b. c.



d.

14.

a.

x

(2, 4) and (2, 14) (-3, 9), (7, 9), (5, 5) (5, 13), (6, 6), (6, 12), (-2, 6), (-2, 12), (-1, 5), (-1, 13) are points with integer coefficients. (x – 2)2 + (y – 9)2 = 25 y

16 14 12 10 8 6 4 2 0 -4 -2 0 2 4 6 8 10 12 14 16 18 20 22 24 -2



b. c.

6 units (x – 17)2 + (y – 9)2 = 36

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x

Practice Answer Keys

Geometry Practice

Practice Answer Keys (cont.) 15.

a.

y

20 18 16 14 12 10 8 6 4 2 0 -4 -2 0 2 4 6 8 10 12 14 16 18 20 22 24 -2

x



b. c.

5 units (x – 8)2 + (y – 13)2 = 25

16.

a.



b.

Answer will vary, but the diameter must be 10. There are several possible endpoints for the diameter, two of which are (0, 2) or (10, 12). 25π units2 ≈ 78.5 square units

17.

a.

y 10 8 6 4 2 0 -4 -2 0 2 4 6 8 10 12 14 16 18 20 22 24 -2

b. c. d.

16π2 units ≈ 50.24 squared units 8π units ≈ 25.12 units 4 2 units

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x

Practice Answer Keys

Geometry Practice

Practice Answer Keys (cont.) 18.

a.



b. c.

19.

a.



b. c.

20.

a.



b. c.



Location of angle may vary, but the vertex should be at the center of a circle and measure 70°. Size of the circle does not matter. 70° 35 π cm 18 Location of angle may vary, but the vertex should be at the center of a circle and measure 50°. Size of the circle does not matter. 50° 25 π cm 18 Location of angle may vary, but the vertex should be on the circumference of a circle and measure 70°. Size of the circle does not matter. The measure of the intercepted arc is 140°. 35 π cm 9

21. a. b.

24π cm ≈ 75.36 cm 144π cm2 ≈ 452.16 cm2

22. a. b.

5π cm ≈ 15.7 cm 6.25π cm2 ≈ 19.625 cm2

23.

20 (0.5) cm ≈ 3.18 cm π

24.

Center is (-2, 1) and radius is 5

25.

a. b.

45° All angles should measure 45° and intercept the same arc.

Chapter 7: Challenge Problems: Circles (page 87) 1.

Answers will vary, but vertex of the angle is at the center of the circle and angle measures 60°.

2.

60°

3.

10 π cm 3

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Practice/Check-Up Answer Keys

Geometry Practice

Practice Answer Keys (cont.) 4.

9π cm ≈ 28.26 cm

5.

20.25π cm2 ≈ 63.585 cm2

6.

1.2π cm ≈ 3.768 cm

7.

0.36π cm2 ≈ 1.1304 cm2

8.

Center is (-5, 6) and radius is 51 units.

9.

x 2 + y 2 = 49

10.

15 π cm π

Check-Up Answer Keys Check-Up Problems: Coordinate Geometry (page 88) 1.

Midpoint of BC =

1 , 3 + -2 = 6 , 1 = (3, 0.5) ( 5 +2 2 2 2 ) ( )

2.

Midpoint of AD =

5 , -2 + 12 = 9 , 10 = (4.5, 5 ) ( 4 +2 2 2 2 ) ( )

3.

Length of DE: 7

4.

Length of AE: 24

5.

Slope = -

6.

!d Slope = - !d

7.

Parallel; The slope of the two lines is the same.

8.

3y – 6x = 18 3y = 6x + 18 y = 2x + 6 Slope = 2

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11 10 9 8 7 6 5 4 3 2 1 0 -11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 -11

(1, 8)

118

(0, 6)

Check-Up Answer Keys

Geometry Practice

Check-Up Answer Keys (cont.) 9.

y = mx + b 0 = 5(-3) + b 15 = b



y = 5x +15

10.

y=7–5=2 x=6–3=3



Slope =



y = mx + b



5 = (3) + b



5=2+b 3=b

y= x+3

11.

(5 – 1)2 + (16 – 4)2



(4)2 + (12)2



16 + 144 = 160 = 4 10

@d

@d

@d

22 20 18 16 14 12 10 8 6 4 2 0 -11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 -2 -4 -6 -8 -10 -12 -14 -16 -18 -20 -22

(1, 20)

(0, 15)

11 10 9 8 7 6 5 4 3 2 1 0 -11-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10 11 -1 -2 -3 -4 -5 -6 -7 -8 -9 -10 -11

(6, 7) (3, 5) (0, 3)

Check-Up Problems: Measuring and Classifying Angles (page 89) All drawings will vary. 1. Angle should be 90°. 2.

Angle should be greater than 90° and less than 180°.

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Geometry Practice

Check-Up Answer Keys

Check-Up Answer Keys (cont.) 3.

Angle should be less than 90°.

4.

Angle should be greater than 180° and less than 360°.

5.

Labels include vertex: ∠A, number: ∠1, three points: ∠BAC and ∠CAB.

6.

Angles must add up to 180°.

7.

Angles must add up to 90°.

8.

Angles 1 and 3 must be equal and 2 and 4 must be equal.

9.

Angle must be divided equally by the bisector.

10.

Angle must be a straight line, 180°.

11.

Angles must be equal.

12.

The angles must share a common side and not have any common interior points.

Check-Up Problems: Triangles (page 90) 1.

A

2.

N

3.

T

4.

T

5.

145 cm

6.

6 145 cm2

7.

P = (8, 23); (5, 14)

8.

a = 4 and b = 7

9.

The perimeter will be greater than 36 cm and will be less than 46 cm.

10.

Answers will vary.

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Geometry Practice

Check-Up Answer Keys

Check-Up Answer Keys (cont.) Check-Up Problems: Polygons and Quadrilaterals (page 91) 1.

A

2.

A

3.

S

4.

T

5.

T

6.

30 cm

7.

The measures of angles B, C, and D are 116°, 64°, and 116°, respectively.

8.

161°

9.

160°

10.

4,140°

Check-Up Problems: Circles (page 92) 1.

Answers will vary, but vertex is on the circle and the angle measures 60°.

2.

120°

3. 4.

20 π cm 3

5.

1.21π cm2 ≈ 3.7994 cm2

6.

44π cm ≈ 138.16 cm

7.

484π cm2 ≈ 1,519.76 cm2

8.

Center is (2, -6), and the radius is 51 units.

9.

x 2 + y 2 = 12.25

10.

288 (0.5) cm π

2.2π cm ≈ 6.908 cm

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References

References Brown, R., Dolciani, M., Sorgenfrey, R., Cole, W., (1997). Algebra Structure and Method Book 1. Evanston, IL: McDougal Littell. Chicago Mathematics Project. Connected Mathematics. University of Chicago. Found online at: http:// www.math.msu.edu/cmp/curriculum/algebra.htm Edwards, E. (1990). Algebra for Everyone. Reston, VA: National Council of Teachers of Mathematics. Freudenthal Institute at the University of Utrecht/University of Wisconsin / NSF. Math in Context. http:// showmecenter.missouri.edu/showme/mic.shtml Larson, R., Boswell, L., Stiff, L. (1998) Geometry and Integrated Approach. Evanston. IL: McDougal Littell. Long, L. (2001). Painless Geometry. Hauppauge, NY: Barron’s Educational Series. National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics. National Council of Teachers of Mathematics (NCTM). (2004). Standards and expectations for algebra. Reston, VA: National Council of Teachers of Mathematics. Found online at: http://www.nctm. org

Web Resources About Math http://math.about.com/ A History of Pi http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/PI_through_the_ages.html Annenburg CPB – Shape and Space in Geometry http://www.learner.org/teacherslab/math/geometry/ Ask Dr. Math http://forum.swarthmore.edu/dr.math/ Binary Numbers http://www.math.grin.edu/~rebelsky/Courses/152/97F/Readings/student-binary.html Boxer Math http://www.boxermath.com Color Math Pink http://www.colormathpink.com

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Geometry Practice

References

References (cont.) Condensed Matter Physics – Quasicrystals http://www.cmp.catech.edu/~lifshitz/symmetry.html Cool Math Sites http://www.cte.jhu.edu/techacademy/web/2000/heal/mathsites.htm Dave’s Math Tables http://www.sisweb.com Designer Factions http://www.math.rice.edu/~Ianius/Patterns/design2.html Ed Helper.com http://www.edhelper.com/algebra.htm Fascinating Folds http://www.education.gov.lc/mathjocv/html%20file/origami/Welcome%20to%20Fascinating%20 folds Figures and Polygons – Math League http://www.mathleague.com/help/geometry/polygons.htm Four Types of Symmetry http://www.mathforum.org/sum95suzanne/symsusan.html Gallery of Interactive Geometry http://www.geom.uiuc.edu/apps/gallery/html Geometric Junkyard http://www.ics.uci.edu/~eppstein/junkyard/topic.html Geometry Junkyard – Symmetry http://www.ics.uci.edu/~eppstein/junkyard/sym.html Geometry Online http://www.math.rice.edu/~Ianius/Geom/ Getting to Know Shapes http://www.illuminations.nctm.org/index_d.aspx?id=406 History of Quilt Patterns http://www.womenfolk.com/quilt_pattern_history/patternlinks.htm Holt, Rinehart, and Winston. Mathematics in Context http://www.hrw.com/math/mathincontext/index.htm

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References

Geometry Practice

References (cont.) Honey Bee Facts http://www.honey.com/kids/facts.html Interactive Mathematic Miscellany and Puzzles http://www.cut-the-knot.org/algebra.shtml Line Symmetry http://www.adrianbruce.com/Symmetry/ Math Archives: Topics in Mathematics http://www.archives.math.utk.edu/topics/ Math for Morons Like Us http://www.library.thinkquest.org/20991/geo/index.html Math Forum http://www.forum.swarthmore.edu/ Math Forum – Geometry/Shapes 1, 2, 3 Dimensions/Vectors http://www.mathforum.org/library/view/2650.html Mega Mathematics http://www.c3.lanl.ov/mega-math/ Miss Glosser’s Math Goodies http://www.mathgoodies.com Oldest Escher Collection on the Web http://www.comcast.net/~davemc0/Escher/ Ole Miss: Problems of the Week http://www.olemmiss.edumathed/problem.htm Patterns Possibilities http://www.cte.jhu.edu/techacademy/web/2000/heal/rsrchist.htm PBS Teacher Source – Geometry and Shapes http://www.pbs.org/teachersource/recommended/mathlk_geometry.shtm Purple Math http://www.purplemath.com/modules/solvrtnl.htm Reflectional Symmetry http://www.geom.uiuc.edu/~demo5337/s97a/reflect.html Regular Polyhedra or Platonic Solids http://www.enchantedlearning.com/math/geometry/solids/ © Mark Twain Media, Inc., Publishers

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References

Geometry Practice

References (cont.) Reichman, H. and Kohn, M.(2004) Math made Easy. Found online at: http://www.mathmadeeasy.com/ Rotational Symmetry http://www.geom.uiuc.edu/~demo5337/s97a/reflect.html Science Behind Snowflakes http://www.msnbc.msn.com/id/3077345/#BODY Science U – Geometry - Symmetry http://www.scienceu.com/geometry/articles/tiling/symmetry.html Science U – Geometry Center http://www.scienceu.com/geometry/ Show Me Center http://www.shoemecenter.missouri.edu/showme/ Snowflakes http://www.snowflakebentley.com/ SOS Mathematics http://www.sosmath.com/ Surfing the Net With Kids http://www.surfnetkids.com/ Symmetry and Patterns of Oriental Rugs http://www.mathforum.org/geometry/rugs/ Symmetry and the Shape of Space http://www.comp.uark.edu/~strauss/symmetry.unit/index.html The Mathematical Art of M.C. Escher http://www.mathacademy.com/pr/minitext/escher/ Think Quest Geometry http://www.library.thinkquest.org/2647/geometry/geometry.htm Totally Tessellations http://www.thinkquest.org/library/site_sum.html?16661&url=16661/escher.html Translational Symmetry http://www.geom.uiuc.edu/~demo5337/s97a/translate.html Two-Dimensional Shapes and Line Symmetry http://www.adrianbruce.com/Symmetry/9.htm © Mark Twain Media, Inc., Publishers

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References

Geometry Practice

References (cont.) University of Akron Theoretical and Applied Mathematics http://www.math.uakron.edu~dpstory/mpt_home.html Wikipedia – Symmetry http://www.en.wikipedia.org/wiki/Symmetry World’s Top Websites on Symmetry http://www.dirs.org/wiki-article-tab.cfm/symmetry

Real Life Applications of Math Applied Academics: Applications of Mathematics – Careers http://www.bced.gov.bc.ca/careers/aa/lessons/math.htm Exactly How Is Math Used in Technology? http://www.math.bcit.ca/examples/index.shtml Geometry in Action http://www.ics.uci.edu/~eppstein/geom.html Line Symmetry http://www.adrianbruce.com/Symmetry/ Mathematics Association of America – Careers http://www.maa.org/careers/index.html NASA Space Link http://www.spacelink.msfc.nasa.gov/index.html Recreational Math http://www.ics.edu/~eppstein/recmath.html

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Look for these Mark Twain Media books for grades 4–8+ at your local teacher bookstore.

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SCIENCE

CD-1807 CD-1808 CD-1809 CD-1810 CD-1811 CD-1815 CD-1816 CD-1817 CD-1818 CD-1820 CD-1304 CD-1305 CD-1315 CD-1319 CD-1327 CD-1338 CD-1387 CD-1534 CD-1535 CD-1536 CD-1537 CD-1538 CD-1552 CD-1558 CD-1559 CD-1564 CD-1565 CD-1568 CD-1577 CD-1613 CD-1614 CD-1631 CD-404005 CD-404006 CD-404007 CD-404024 CD-404025 CD-404034 CD-404045 CD-404046 CD-404047 CD-404048 CD-404048 CD-404050 CD-404052

Geology: Our Planet Earth Rocks, Minerals, & Fossils Meteorology Terrestrial Biomes Earthquakes & Volcanoes Science Experiments: Chem./Physics Science Experiments: Earth Science Science Experiments: Biology/Ecology The Atom Physical Science Plants Resourceful Rain Forest Microorganisms Science Skills Made Easy Science Fair Projects Your Body and How It Works Elements and the Periodic Table Learning About Amphibians Learning About Birds Learning About Fishes Learning About Mammals Learning About Reptiles Learning About Invertebrates Simple Machines Chemistry Atmosphere and Weather Rocks and Minerals Electricity and Magnetism Learning About DNA Light and Color Sound Learning About Atoms The Solar System Learning About Vertebrates Learning About Our Solar System Jumpstarters for Science Science Tutor: Chemistry Science Tutor: Life Science Science Tutor: Physical Science Science Tutor: Earth & Space Science Easy Science Experiments: Weather Easy Science Experiments: The Earth's Surface Easy Science Experiments: Water, Airplanes, … Learning About Cells Amazing Facts About Mammals

CD-1827 CD-1828 CD-1829 CD-1830 CD-1831 CD-1832 CD-1835 CD-1837 CD-1839 CD-1860 CD-1885 CD-1886 CD-1887 CD-1888 CD-1873 CD-1899 CD-1301 CD-1302 CD-1309 CD-1311 CD-1312 CD-1316 CD-1317 CD-1318 CD-1323 CD-1326 CD-1330 CD-1335 CD-1336 CD-1360 CD-1361 CD-1367 CD-1385

American Women Civil War: The War Between the States Greek and Roman Mythology Medieval Times: 325-1453 Understanding the U.S. Constitution Explorers of the New World World War II The Industrial Revolution Egypt and the Middle East Democracy, Law, and Justice Economics and You Mayan, Incan, and Aztec Civilizations The American Revolution Greek and Roman Civilizations Seven Wonders of the World and More Holocaust South America Renaissance Elections Canada Heroes Africa Disasters Basic Economics Mexico Personal Finance Social Studies Skills Made Easy Rivers of the U.S. U.S. History Maps U.S. Constitution: Preparing for the Test 50 U.S. States and Territories World Civilizations and Cultures Amazing Facts in U.S. History

SOCIAL STUDIES

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CD-1390 CD-1392 CD-1395 CD-1396 CD-1397 CD-1525 CD-1526 CD-1527 CD-1528 CD-1529 CD-1530 CD-1531 CD-1532 CD-1533 CD-1550 CD-1556 CD-1560 CD-1561 CD-1562 CD-1563 CD-1572 CD-1584 CD-404026 CD-404031 CD-404032 CD-404033 CD-404036 CD-404037 CD-404038 CD-404039 CD-404040

U.S. Presidents: Past and Present Constitutional Puzzlers Discovering and Exploring the Americas Life in the Colonies The American Revolution The California Gold Rush The Lewis and Clark Expedition The Oregon and Santa Fe Trails The Westward Movement Abraham Lincoln and His Times Industrialization in America Slavery in the United States The American Civil War The Reconstruction Era We the People: Government in America The RoaringTwenties/Great Depression Coming to America: Immigration America in the 1960s and 1970s America in the 1980s and 1990s World War II and the Post-War Years Understanding Investment/Stock Market Amazing Facts in World History Jumpstarters for U.S. History Jumpstarters for the U.S. Constitution Wonders: A Journey Around the World Mysteries: A Journey Around the World U.S. History: People Who Helped Make the Republic Great: 1620–Present U.S. History: Inventors, Scientists, Artists, & Authors U.S. History: People and Events in African-American History U.S. History: People and Events: 1607–1865 U.S. History: People and Events: 1865–Present

GEOGRAPHY

CD-1314 Map Reading, Latitude, Longitude, … CD-1551 World Geography CD-1555 Exploring Asia CD-1556 Exploring Africa CD-1566 Exploring Europe CD-1567 Exploring South America CD-1569 Exploring North America CD-1570 Exploring Antarctica CD-1571 Exploring Australia CD-1573–CD-1576 Discovering the World of Geography: Grades 4–8

LANGUAGE ARTS

CD-1850 How to Prepare and Give a Speech CD-1851 Writing CD-1857 Grammar and Composition CD-1862 Writing to Inform and Persuade CD-1300 Phonics for Middle-Grade Students CD-1320 Reading Skills Made Easy CD-1366 Debate Skills CD-1372–CD-1375 Reading Comprehension: G4-7 CD-1381 Confusing Words CD-1382 Synonyms and Antonyms CD-1383 Using a Dictionary CD-1384 Word Pronunciation CD-1386 Challenges Galore:Vocabulary Building CD-1391 Challenges Galore: Grades 4-5 CD-1393 Proofreading CD-1394 Story Writing CD-1398 Making Classroom Videos CD-1399 Poetry Writing CD-1543–CD-1546, CD-1553 Writing Engagement: Grades 4–8 CD-1547 Essential Words CD-1549 Report and Term Paper Writing CD-1554 English Warm-ups CD-1585–CD-1588 Ready, Set, Write: Story Starters: Grades 4–7 CD-1590–CD-1595 Student Booster Writing series CD-1620–CD-1624 Reading Tutor series CD-404008 Diagraming Sentences CD-404011 Jumpstarters for Grammar CD-404012 L.A. Tutor: Capitalization/Punctuation CD-404013 Language Arts Tutor: Grammar

CD-404015–CD-404019 Reading Engagement: Grades 3–8 CD-404027 Jumpstarters for Writing * CD-404035 Lessons in Writing * CD-404051 Writing a Persuasive Essay

STUDY SKILLS

CD-1859 Improving Study & Test-Taking Skills CD-1898 Listening Skills CD-1321 Library Skills CD-1597 Note Taking: Lessons to Improve Research Skills & Test Scores CD-1598 Developing Creative Thinking Skills CD-1625–CD-1630 Preparing Students for Standardized Testing: Grades 3–8

MATH

CD-1874 Algebra CD-1876 Pre-Algebra CD-1877 Pre-Geometry CD-1878 Math Journal CD-1879 Statistics and Probability CD-1310 Understanding Graphs & Charts CD-1325 Pre-Calculus CD-1329 Word Problems CD-1331 Applying Pre-Algebra CD-1332 Basic Geometry CD-1333 Fractions, Decimals, and Percentages CD-1388 Math-O Games CD-1389 Math Warmups CD-1539–CD-1542, CD-1589 Math Twisters: Gr. 4–8 CD-1578–CD-1582 Math Engagement: Gr. 4–8 CD-1589 Math Projects CD-1615–CD-1619 Math Tutor series CD-404000–404004 Math Games: Gr. 4–8 CD-404009 Math Challenges CD-404020 Helping Students Understand Algebra CD-404021 Helping Students Understand Pre-Algebra CD-404022 Jumpstarters for Algebra CD-404023 Jumpstarters for Math CD-404028 Helping Students Understand Algebra II CD-404029 Helping Students Understand Geometry CD-404030 Jumpstarters for Pre-Algebra * CD-404041 Pre-Algebra Practice * CD-404042 Algebra Practice * CD-404043 Algebra II Practice * CD-404044 Geometry Practice

FINE ARTS

Music: a.d. 450–1995 Great Artists and Musicians American Popular Music Theater Through the Ages Music of Many Cultures Musical Instruments of the World Everyday Art for the Classroom

CD-1890 CD-1891 CD-1892 CD-1893 CD-1894 CD-1596 CD-1632

CD-1895 CD-1896 CD-1897 CD-1819 CD-1339

CD-1363 CD-1364 CD-1365 CD-1369 CD-1370 CD-1371 CD-1548

HEALTH & WELL-BEING

Life Skills Self-Management: Promoting Success Promoting Positive Values Health, Wellness, and Physical Fitness Developing Life Skills

CURRICULUM PUZZLES Spelling Puzzles Grades 4-5 Spelling Puzzles Grades 5-6 U.S. Presidents Puzzles Mathematics Puzzles Spelling Puzzles Grades 6-7 U.S. History Puzzles Music & Art History Puzzles

CROSS-CURRICULUM

CD-1306 CD-1307 CD-1313 CD-404010

Career Search Classroom Publishing Using the Internet Professional Teacher Calendar *Denotes New Release