7A Two-Dimensional Geometry 7-1 Points, Lines, Planes, and Angles LAB Bisect Figures 7-2 Parallel and Perpendicular Lines LAB Constructions 7-3 Angles in Triangles 7-4 Classifying Polygons LAB Exterior Angles of a Polygon 7-5 Coordinate Geometry 7B Patterns in Geometry 7-6 Congruence 7-7 Transformations LAB Combine Transformations 7-8 Symmetry 7-9 Tessellations Playground Equipment Designer Playground equipment must be attractive, safe, fun, and appropriate for the ages of the children who will use it. Years ago, designers used pencils, T-squares, and slide rules to create their designs. Designers now use computers, 3-D programs, and virtual reality to design playgrounds. Foundations of Geometry Foundations of Geometry 320 Chapter 7 KEYWORD: MT7 Ch7 Shapes of Playground Equipment Equipment Ground Shape Merry-go-round Circle Four-square court Square Swings Rectangle Climbing structure Octagon
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7A Two-DimensionalGeometry
7-1 Points, Lines, Planes,and Angles
LAB Bisect Figures
7-2 Parallel and Perpendicular Lines
LAB Constructions
7-3 Angles in Triangles
7-4 Classifying Polygons
LAB Exterior Angles of aPolygon
7-5 Coordinate Geometry
7B Patterns in Geometry7-6 Congruence
7-7 Transformations
LAB Combine Transformations
7-8 Symmetry
7-9 Tessellations
PlaygroundEquipment Designer
Playground equipmentmust be attractive, safe, fun,and appropriate for the agesof the children who will useit. Years ago, designers usedpencils, T-squares, and sliderules to create their designs.Designers now use computers,3-D programs, and virtual reality to design playgrounds.
Foundations ofGeometryFoundations ofGeometry
320 Chapter 7
KEYWORD: MT7 Ch7
Shapes of Playground Equipment
EquipmentGround Shape
Merry-go-round Circle
Four-square court Square
SwingsRectangle
Climbing structure Octagon
m807_c07_320_321 1/13/06 11:45 AM Page 320
Foundations of Geometry 321
VocabularyChoose the best term from the list to complete each sentence.
1. In the __?__ (4, �3), 4 is the __?__, and �3 is the __?__.
2. The __?__ divide the __?__ into four sections.
3. The point (0, 0) is called the __?__.
4. The point (0, �3) lies on the __?__, while the point (�2, 0) lies on the __?__.
Complete these exercises to review skills you will need for this chapter.
Ordered Pairs Write the coordinates of the indicated points.
5. point A 6. point B
7. point C 8. point D
9. point E 10. point F
11. point G 12. point H
Similar FiguresTell whether the figures in each pair appear to be similar.
Determine whether the given values are solutions of the given equations.
23. �23�x � 1 � 7 x � 9 24. 2x � 4 � 6 x � �1
25. 8 � 2x � �4 x � 5 26. �12�x � 5 � �2 x � �14
O 4
4
2
2�2�4
�2
�4
x
y
A
B
CD E
F
G H
coordinate axes
coordinate plane
ordered pair
origin
x-axis
x-coordinate
y-axis
y-coordinate
m807_c07_320_321 1/13/06 11:45 AM Page 321
Previously, you
• located and named points on a coordinate plane.
• recognized geometric conceptsand properties in fields such asart and architecture.
• used critical attributes to define similarity.
You will study
• graphing translations andreflections on a coordinateplane.
• using geometric concepts andproperties of geometry to solveproblems in fields such as artand architecture.
• using critical attributes todefine congruency.
You can use the skillslearned in this chapter
• to find angle measures by usingrelationships within figures.
• to create tessellations.
• to identify properties ofgeometry in art andarchitecture.
Vocabulary ConnectionsTo become familiar with some of thevocabulary terms in the chapter, consider thefollowing. You may refer to the chapter, theglossary, or a dictionary if you like.
1. The word equilateral contains the rootsequi, which means “equal,” and lateral,which means “of the side.” What do yousuppose an is?
2. The Greek prefix poly means “many,” andthe root gon means “angle.” What do yousuppose a is?
3. Think of what means when you aretalking about a hill. How do you think thisapplies to lines on a coordinate plane?
I’m having trouble with Lesson 6-5. I can find what
percent one number is of another number, but I get
confused about finding percent increase and decrease.
My teacher helped me think it through:
Find the percent increase or decrease from 20 to 25.
• First figure out if it is a percent increase or decrease.
It goes from a smaller to a larger number, so it is
a percent increase because the number is getting larger,
or increasing.
• Then find the amount of increase, or the difference,
between the two numbers. 25 � 20 � 5
• Now find what percent the amount of increase, or
difference, is of the original number.
�am
o
o
r
u
ig
n
i
t
na
o
l
f
n
i
u
n
m
cr
b
e
e
a
r
se� → �
2
5
0� � 0.25 � 25%
So it is a 25% increase.
Writing Strategy: Keep a Math JournalBy keeping a math journal, you can improve your writing and thinkingskills. Use your journal to summarize key ideas and vocabulary from eachlesson and to analyze any questions you may have about a concept oryour homework.
Journal Entry: Read the entry a student made in her journal.
Begin a math journal. Write in it each day this week, using theseideas as starters. Be sure to date and number each page.
■ In this lesson, I already know . . .
■ In this lesson, I am unsure about . . .
■ The skills I need to complete this lesson are . . .
■ The challenges I encountered were . . .
■ I handled these challenges by . . .
■ In this lesson, I enjoyed/did not enjoy . . .
Try This
Read
ing
and
Writin
g M
ath
Reading and Writing Math 323
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Learn to classify andname figures.
Vocabulary
vertical angles
congruent
supplementaryangles
complementaryangles
obtuse angle
acute angle
right angle
angleray
segmentplane
linepoint
Points, lines, and planes are the building blocks of geometry.Segments, rays, and angles are defined in terms of these basic figures.
GHG
H
A names a location. • A point A
A is perfectly straight and extends forever in both directions.
A is a perfectly flat surface thatextends forever in all directions.
A , or line segment, is the part of a line between two points.
A is part of a line that starts at one point and extends forever in one direction.
ray
segment
plane
line
point
Z
Q R S
T
J
K
KJ
BC��� is read “line BC.” GH��� is read “segment GH.” KJ�� is read “ray KJ.” Toname a ray, always write the endpoint first.
Naming Points, Lines, Planes, Segments, and Rays
Use the diagram to name each figure.
four points
Q, R, S, T
a line Any 2 points on the line can
Possible answers: QS���, QR��� or RS��� be used.
a plane
Possible answers: Any 3 points in the plane that plane Z or plane QRT form a triangle can name a plane.
four segments Write the 2 points in any order,
Possible answers: QR���, RS���, RT���, QS��� for example, Q�R� or R�Q�.
five rays
RQ��, RS��, RT��, SQ��, QS�� Write the endpoint first.
E X A M P L E 1
line �, or BC
BC
�
EPplane P, orplane DEFF
D
324 Chapter 7 Foundations of Geometry
7-1 Points, Lines, Planes,and Angles
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A measures 90°. An measures greater than 0° and
less than 90°. An measures greater than 90° and less than
• Segments that have the same length are congruent.
• Angles that have the same measure are congruent.
• The symbol for congruence is �, which is read “is congruent to.”
Intersecting lines form two pairs of . Vertical angles arealways congruent, as shown in the next example.
vertical angles
Congruent
Supplementary angles
Complementary angles
obtuse angle
acute angleright angle
The measures of angles that fittogether to form a straight line,such as �FKG, �GKH, and�HKJ, add to 180°.
The measures of angles that fittogether to form a completecircle, such as �MRN, �NRP,�PRQ, and �QRM, add to 360°.
An (�) is formed by two rays with a common endpoint called the vertex (plural, vertices). Angles can be measured in degrees. m�1 means the measure of �1. The angle can be named �XYZ, �ZYX, �1, or �Y. The vertex must be the middle letter.
angle
Y
1Z
X
F
GH
K JR
M
NP
Q
A
C
30°90°
60°
D
E BA right angle can belabeled with a smallbox at the vertex.
2E X A M P L E
7-1 Points, Lines, Planes, and Angles 325
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7-1 ExercisesExercises
Use the diagram to name each figure.
1. three points 2. a line 3. a plane
4. three segments 5. three rays
Use the diagram to name each figure.
6. a right angle 7. two acute angles
8. an obtuse angle 9. a pair of complementary angles
10. two pairs of supplementary angles
KEYWORD: MT7 Parent
KEYWORD: MT7 7-1
GUIDED PRACTICE
See Example 2
See Example 1
Think and Discuss
1. Tell which statements are correct if �X and �Y are congruent.
a. �X � �Y b. m�X � m�Y c. �X � �Y d. m�X � m�Y
2. Explain why vertical angles must always be congruent.
Finding the Measures of Vertical Angles
In the figure, �1 and �3 are vertical angles, and�2 and �4 are vertical angles.
In the figure, �1 and �3 are vertical angles,and �2 and �4 are vertical angles.
11. If m�3 � 105°, find m�1.
12. If m�2 � x°, find m�4.
Use the diagram to name each figure.
13. four points 14. two lines
15. a plane 16. three segments
17. five rays
Use the diagram to name each figure.
18. a right angle
19. two acute angles
20. two obtuse angles
21. a pair of complementary angles
22. two pairs of supplementary angles
In the figure, �1 and �3 are vertical angles,and �2 and �4 are vertical angles.
23. If m�2 � 126°, find m�4.
24. If m�1 � b°, find m�3.
Use the figure for Exercises 25–34. Write true or false. If a statement is false,rewrite it so it is true.
25. NQ��� is a line in the figure.
26. Rays UQ�� and UT�� make up line TQ���.
27. �QUR is an obtuse angle.
28. �4 and �2 are supplementary.
29. �1 and �6 are supplementary.
30. �3 and �1 are complementary.
31. If m�1 � 35°, then m�6 � 40°.
32. If m�SUN � 150°, then m�SUR � 150°.
33. If m�1 � x° , then m�PUQ � 180° � x° .
34. m�1 � m�3 � m�5 � m�6 � 180°.
35. Critical Thinking Two complementary angles have a ratio of 1:2. What is the measure of each angle?
PRACTICE AND PROBLEM SOLVING
INDEPENDENT PRACTICE
Extra PracticeSee page 794.
See Example 3
See Example 3
See Example 2
See Example 1
12
34
12
34
NJ
L
M
K
VWX
YZ
60°30°
Q
U
T
N
R
P
S
61
35
2
4
7-1 Points, Lines, Planes, and Angles 327
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Physical Science
40. Multiple Choice When two angles are complementary, what is the sum of their measures?
90° 180° 270° 360°
41. Gridded Response �1 and �3 are supplementary angles. If m�1 � 63°, find m�3.
Multiply. Write the product as one power. (Lesson 4-3)
42. m3 � m2 43. w � w6 44. 78 � 73 45. 116 � 119
46. Callie made a 5 in. tall by 7 in. wide postcard. A company would like to sell a poster based on the postcard. The poster will be 2 ft tall. How wide will the poster be? (Lesson 5-5)
DCBA
Physical ScienceThe archerfish can spit a stream of water up to 3 meters inthe air to knock its prey into the water. This job is mademore difficult by refraction, the bending of light waves asthey pass from one substance to another. When you lookat an object through water, the light between you and theobject is refracted. Refraction makes the object appear tobe in a different location. Despite refraction, the archerfishstill catches its prey.
36. Suppose that the measure of the angle between the bug’sactual location and the bug’s apparent location is 35°.
a. Refer to the diagram. Along the fish’s line of vision,what is the measure of the angle between the fish andthe bug’s apparent location?
b. What is the relationship of the angles in the diagram?
37. In the image, the underwater part of the net appears to be 40° to the right of where it actually is. What is themeasure of the angle formed by the image of theunderwater part of the net and the part of the net above the water?
38. Write About It Suppose an archerfish is directly below its prey.Explain why there would be little or no distortion.
39. Challenge A person on the shore is looking at a fish in the water. At the same time, the fish is looking at the person from below the surface.Describe what each observer sees, and where the person and the fishactually are in relation to where they appear to be.
328 Chapter 7 Foundations of Geometry
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Follow the steps below to bisect a segment.
a. Draw J�K� on your paper. Place your compass point on Jand draw an arc. Without changing your compass opening, place your compass point on K and draw an arc.
b. Connect the intersections of the arcs with a line. MeasureJ�M� and K�M�. What do you notice?
Follow the steps below to bisect an angle.
a. Draw acute �H on your paper. b. Place your compass point on H and draw an arc through both sides of the angle.
c. Without changing your compass d. Draw HD��. Measure �GHD and �DHE.opening, draw intersecting arcs from What do you notice?G and E. Label the intersection D.
1. Explain how to use a compass and a straightedge to divide a segmentinto four congruent segments. Prove that the segments are congruent.
Draw each figure, and then use a compass and a straightedge to bisect it.Verify by measuring.
1. a 2-inch segment 2. a 0.5-inch segment 3. a 6-inch segment
4. a 48° angle 5. a 90° angle 6. a 110° angle
Bisect Figures
Use with Lesson 7-1
7-1
Think and Discuss
Try This
KEYWORD: MT7 Lab7When you bisect a figure, you divide it into two congruent parts.
J K
J KM
1
E
D
H
G
E
DH
G
H
EH
G
2
Activity
7-1 Hands-On Lab 329
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Learn to identifyparallel and perpendicular lines and the angles formed by a transversal.
Vocabulary
transversal
perpendicular lines
parallel lines
are lines in aplane that never meet, suchas the opposite edges of askyscraper’s windows. Theedges appear to get closerto each other because ofperspective.
An edge and the bottom of a window are like
; that is,they intersect at 90° angles.
A is a line that intersects two or more lines that lie in the sameplane. Transversals to parallel lines form angles with special properties.
Identifying Congruent Angles Formed by a Transversal
Copy and measure the anglesformed by the transversal andthe parallel lines. Which anglesseem to be congruent?
�1, �4, �5, and �8 all measure 60°.�2, �3, �6, and �7 all measure 120°.
Angles marked in blue appearcongruent to each other, andangles marked in red appearcongruent to each other. �1 � �4 � �5 � �8�2 � �3 � �6 � �7
transversal
perpendicular lines
Parallel lines
You cannot tell ifangles are congruentby measuring becausemeasurement is notexact.
The sides of the windows are transversals to the top and bottom.
The top andbottom of thewindowsare parallel.
1 23 4
5 67 8
1 23 4
5 67 8
E X A M P L E 1
330 Chapter 7 Foundations of Geometry
7-2 Parallel and PerpendicularLines
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Finding Angle Measures of Parallel Lines Cut by Transversals
In the figure, line a⏐⏐line b. Find the measureof each angle.
�4
m�4 � 74� Corresponding angles are congruent.
�3
m�3 � 74° � 180� �3 is supplementary to the 74� angle.
� 74� � 74� Subtract 74� from both sides.����� ���m�3 � 106� Simplify.
�5
m�5 � 106� �3 and �5 are alternate interior angles, sothey are congruent.
Some pairs of the eight angles formed by two parallel lines and atransversal have special names.
PROPERTIES OF TRANSVERSALS TO PARALLEL LINES
If two parallel lines are intersected by a transversal, • corresponding angles are congruent, • alternate interior angles are congruent, • and alternate exterior angles are congruent.
If the transversal is perpendicular to the parallel lines, all of theangles formed are congruent 90� angles.
1. Tell how many different angles would be formed by a transversalintersecting three parallel lines. How many different anglemeasures would there be?
2. Explain how a transversal could intersect two other lines so thatcorresponding angles are not congruent.
2E X A M P L E
The symbol forparallel is⏐⏐. Thesymbol forperpendicular is ⊥.
7-2 Parallel and Perpendicular Lines 331
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7-2 ExercisesExercises
1. Measure the angles formed by thetransversal and the parallel lines.Which angles seem to becongruent?
In the figure, line m⏐⏐line n. Find themeasure of each angle.
2. �1 3. �4
4. �6 5. �7
6. Measure the angles formed by the transversal andthe parallel lines. Which angles seem to becongruent?
In the figure, line p⏐⏐line q. Find the measure of each angle.
7. �1
8. �4
9. �6
10. �7
In the figure, line t⏐⏐line s.
11. Name all angles congruent to �1.
12. Name all angles congruent to �2.
13. Name three pairs of supplementary angles.
14. Which line is the transversal?
15. If m�4 is 51�, what is m�2?
16. If m�7 is 116�, what is m�3?
17. If m�5 is 91�, what is m�2?
Draw a diagram to illustrate each of the following.
18. line p⏐⏐line q⏐⏐line r and line s transversal to lines p, q, and r
19. line m⏐⏐line n and transversal h with congruent angles �1 and �3
20. line h⏐⏐line j and transversal k with eight congruent angles
KEYWORD: MT7 Parent
KEYWORD: MT7 7-2
GUIDED PRACTICE
PRACTICE AND PROBLEM SOLVING
INDEPENDENT PRACTICE
Extra PracticeSee page 794.
See Example 2
See Example 1
See Example 2
See Example 1
1 2 5 63 4 7 8
1
4 237
68
5
t
r
s
1 42
37
65
p q
110°
12
34
5 86
7
123
n
476
5
m
t
118°
332 Chapter 7 Foundations of Geometry
m807_c07_330_333 1/13/06 11:46 AM Page 332
A
B
21. Critical Thinking Two parallel lines are cut by a transversal. Can youdetermine the measures of all the angles formed if given only one anglemeasure? Explain.
22. Physical Science A periscopecontains two parallel mirrors thatface each other. With a periscope, aperson in a submerged submarinecan see above the surface of thewater.
a. Name the transversal in the diagram.
b. If m�1 � 45°, find m�2, m�3, and m�4.
23. What’s the Error? Line a is parallel to line b. Line c is perpendicular toline b. Line c forms a 60° angle with line a. Why is this figure impossible todraw?
24. Write About It Choose an example of abstract art or architecture withparallel lines. Explain how parallel lines, transversals, or perpendicular linesare used in the composition.
25. Challenge In the figure, �1, �4, �6, and �7 are all congruent, and �2, �3, �5, and �8 are all congruent. Does this mean that line s is parallel to line t? Explain.
26. Multiple Choice Two parallel lines are intersected by a transversal. The measures of two corresponding angles that are formed are each 54°.What are the measures of each of the angles supplementary to thecorresponding angles?
36° 72° 108° 126°
27. Extended Response Suppose a transversal intersects two parallel lines.One angle that is formed is a right angle. What are the measures of theremaining angles? What is the relationship between the transversal andthe parallel lines?
Find each number. (Lesson 6-3)
28. What is 15% of 96? 29. What is 146% of 12,500?
30. What is 0.5% of 1000? 31. What is 99.9% of 1500?
�1 and �3 are vertical angles, and �2 and �4 are vertical angles. �1 and �2 aresupplementary angles. (Lesson 7-1)
32. If m�1 � 25°, find m�3°. 33. If m�2 � 95°, find m�3.
DCBA
Physical Science
This hat from the 1937 BritishIndustries Fair isequipped with a pair of parallelmirrors to enablethe wearer to seeabove crowds.
KEYWORD: MT7 Periscope
�1 � �2�3 � �4
1
2
34
56 8
7r
s t
7-2 Parallel and Perpendicular Lines 333
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Constructions
Use with Lesson 7-2
7-2
Follow the steps below to construct an angle congruent to �B.
a. Draw acute �ABC on your paper. Draw DE��.
b. With your compass point on B, draw an arc through �ABC. With the same compass opening, place your compass point on D and draw an arc through DE��. Label the intersection point F.
c. Adjust your compass to the width of the arc intersecting �ABC. Place your compass point on F and draw an arc that intersects the arc through DE�� at G. Draw DG��. Measure �ABC and �GDF.
Follow the steps below to construct parallel line segments.
1. Construct Q�R� on your paper by placing the point of your compass on Qand the pencil on R below. Draw point Q on your paper and place the point of your compass on it. Make a short arc and draw a line from Q to the arc. The intersection of the point and the arc is R. Draw point S above or below Q�R�.Draw a line through point S that intersects Q�R�. Label the intersection T.
2. Construct an angle with its vertex at S congruent to �STR. Use themethod described in . How do you know the lines are parallel?
Activity
KEYWORD: MT7 Lab7Constructing an angle is an important step in the construction ofparallel lines.
Q T R
S
Q T R
SW
Q T R
SW
CB
A
ED
CB
A
EFD
CB
A
EFD
G
2
1
334 Chapter 7 Foundations of Geometry
1
Q R Q R
S
m807_c07_334_335 1/13/06 11:47 AM Page 334
Follow the steps below to construct perpendicular lines.
a. Draw MN��� on your paper. Draw point P b. With your compass point at P, draw an arc above or below MN���. intersecting MN��� at points Q and R.
c. Draw arcs from points Q and R, d. Draw PS���. What do you think is true about using the same compass opening, MN��� and PS��� ? Check your guess.that intersect at point S.
1. How many lines can be drawn that are perpendicular to a given line?Explain your answer.
2. Name three ways that you can determine if two lines are parallel.
Use a compass and a straightedge to construct each figure.
1. an angle congruent to �LMN 2. a line parallel to ST��� 3. a line perpendicular to GH���
4. an angle congruent to �DEF 5. a line parallel to AB��� 6. a line perpendicular to CD���
Think and Discuss
Try This
P
M N
Q
P
M N
R
Q
S
P
M N
RQ
S
P
M N
R
L
MN S
T
H
G
FE
D
B
A D
C
3
7-2 Hands-On Lab 335
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Learn to find unknownangles in triangles.
Vocabulary
scalene triangle
isosceles triangle
equilateral triangle
obtuse triangle
right triangle
acute triangle
Triangle Sum Theorem
If you tear off two corners of atriangle and place them next to thethird corner, the three angles seemto form a straight angle.
Draw a triangle and extend one side.Then draw a line parallel to the extended side, as shown.
The three angles in the triangle canbe arranged to form a straight angle, or 180°.
The sides of the triangle are transversals to the parallel lines.
An has 3 acute angles. A has 1 right angle. An has 1 obtuse angle.
Finding Angles in Acute, Right, and Obtuse Triangles
Find x° in the acute triangle.
63° � 42° � x ° � 180° Triangle Sum Theorem
105° � x ° � 180°� 105° � 105° Subtract 105° from ����� ���
This torn triangle demonstrates an importantgeometry theorem called the Triangle Sum Theorem.
m807_c07_336_340 1/13/06 11:53 AM Page 336
Find z° in the obtuse triangle.
13° � 62° � z ° � 180° Triangle Sum Theorem75° � z ° � 180°
� 75° � 75° Subtract 75° from����� ����
z ° � 105° both sides.
An has 3 congruent sides and 3 congruent angles. Anhas at least 2 congruent sides and 2 congruent angles. A
has no congruent sides and no congruent angles.
Finding Angles in Equilateral, Isosceles, and Scalene Triangles
Find the angle measures in the equilateral triangle.
3m° � 180° Triangle Sum Theorem
�3m
3°
� � �18
30°� Divide both sides by 3.
m° � 60°
All three angles measure 60°.
Find the angle measures in the isosceles triangle.
55° � n° � n° � 180° Triangle Sum Theorem
55° � 2n° � 180° Simplify.� 55° � 55° Subtract 55° from both sides.������ ���
2n° � 125°
�2
2n°� � �
1225°� Divide both sides by 2.
n° � 62.5°
The angles labeled n° measure 62.5°.
Find the angle measures in the scalene triangle.
2p° � 3p° � 4p° � 180° Triangle Sum Theorem
9p° � 180° Simplify.
�9
9p°� � �
1890°� Divide both sides by 9.
p° � 20°
The angle labeled 2p° measures 2(20°) � 40°, the angle labeled 3p° measures 3(20°) � 60°, and the angle labeled 4p° measures 4(20°) � 80°.
scalene triangleisosceles triangle
equilateral triangle
2E X A M P L E
13°
62°z°
m°
m° m°
55°
n° n°
4p°
3p° 2p°
7-3 Angles in Triangles 337
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7-3 ExercisesExercises
1. Find q° in the acute triangle.
2. Find r° in the right triangle.
3. Find s° in the obtuse triangle.
4. Find the angle measures in the equilateral triangle.
5. Find the angle measures in the isosceles triangle.
6. Find the angle measures in the scalene triangle.
7. The second angle in a triangle is half as large as the first. The third angle isthree times as large as the second. Find the angle measures and draw apossible picture.
KEYWORD: MT7 Parent
KEYWORD: MT7 7-3
GUIDED PRACTICE
See Example 2
See Example 3
See Example 1
Finding Angles in a Triangle That Meets Given Conditions
The second angle in a triangle is twice as large as the first. Thethird angle is half as large as the second. Find the angle measuresand draw a possible figure.
Let x ° � first angle measure. Then 2x ° � second angle measure, and�12�(2x)° � x ° � third angle measure.
x° � 2x° � x° � 180° Triangle Sum Theorem
�44x°� � �
1840°�
Simplify, then divide bothsides by 4.
x ° � 45°
Two angles measure 45° and one angle measures 90°. The triangle hastwo congruent angles. The triangle is an isosceles right triangle.
Think and Discuss
1. Explain whether a right triangle can be equilateral. Can it beisosceles? scalene?
2. Explain whether a triangle can have 2 right angles. Can it have 2obtuse angles?
45°
15°
s°
c° c°
68°
a°
a°
a°
E X A M P L E 3
31° r°
70° 33°
q°
5d°
d°
4d°
45°
45°
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8. Find r ° in the acute triangle.
9. Find s ° in the right triangle.
10. Find t ° in the obtuse triangle.
11. Find the angle measures in the equilateral triangle.
12. Find the angle measures in the isosceles triangle.
13. Find the angle measures in the scalene triangle.
14. The second angle in a triangle is five times as large as the first. The third angle is two-thirds as large as the first. Find the angle measures and draw a possible picture.
Find the value of each variable.
15. 16. 17.
18. 19. 20.
Sketch a triangle to fit each description. If no triangle can be drawn, write not possible.
21. acute scalene 22. obtuse equilateral 23. right scalene
24. right equilateral 25. obtuse scalene 26. acute isosceles
27. Triangle ABC is a right triangle and m�A � 38°. What does the third angle measure?
28. Can an acute isosceles triangle have two angles that measure 40°? Explain.
29. Triangle LMN is an obtuse triangle and m�L � 25°. �M is the obtuse angle.What is the largest m�N can be to the nearest whole degree?
PRACTICE AND PROBLEM SOLVING
INDEPENDENT PRACTICE
Extra PracticeSee page 794.
See Example 2
See Example 3
See Example 1
w°
w°
w°
44°
x°79°
y°
34° 34°29°
121°
w°
40°
x°
6x°
y°5y° w°
(w + 15)°45°
32°
s°r°
23° 71°
36°
m° m°
25° 40°
t°
7g°2g°
9g°
7-3 Angles in Triangles 339
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30. Social Studies American Samoa is aterritory of the United States made up of agroup of islands in the Pacific Ocean, abouthalfway between Hawaii and New Zealand.The flag of American Samoa is shown.
a. Find the measure of each angle in the blue triangles.
b. Use your answers to part a to find theangle measures in the white triangle.
c. Classify the triangles in the flag by theirsides and angles.
31. Choose a Strategy Which of the following sets of angle measures can be used to create an isosceles triangle?
32. Write About It Explain how to cut a square or an equilateral triangle in half to form two identical triangles. What are the angle measures in theresulting triangles in each case?
33. Challenge Find x, y, and z.
DCBA
34. Multiple Choice Which type of triangle can be constructed with a 50°angle between two 8-inch sides?
Equilateral Isosceles Scalene Obtuse
35. Short Response Two angles of a triangle are 45° and 30°. What is the measure of the third angle? Is the triangle acute, right, or obtuse?
Each square root is between two integers. Name the integers. (Lesson 4-6)
36. �42� 37. �71� 38. �35� 39. �296�
In the figure, line x⏐⏐line y. (Lesson 7-2)
40. If m�1 � 34°, what is m�7?
41. If m�6 � 125°, what is m�5?
42. If m�1 � 34°, what is m�4?
43. If m�5 � 34°, what is m�2?
DCBA
1
78 2
6 43
5
x y
d
y˚
w˚
m˚
z˚
15˚
15˚
x˚
z°x°y°
20°40°
110°110°
340 Chapter 7 Foundations of Geometry
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Learn to classify andfind angles in polygons.
Vocabulary
square
rhombus
rectangle
parallelogram
trapezoid
regular polygon
polygon
E X A M P L E 1
Quadrilateral
Hexagon
Pentagon
Polygon Number of Sides
Triangle 3
Quadrilateral 4
Pentagon 5
Hexagon 6
Heptagon 7
Octagon 8
n-gon n
Hexagon:6 sides4 triangles
Heptagon:7 sides5 triangles
Kites have been around for over3000 years, when the Chinesemade them from bamboo and silk.The most common flat kite is inthe shape of a diamond, a type ofquadrilateral called a kite.
A is a closed plane figure formed by three or moresegments. A polygon is named by the number of its sides.
Finding Sums of the Angle Measures in Polygons
Find the sum of the angle measures in each figure.
Find the sum of the angle measures in a quadrilateral.
Divide the figure into triangles.
2 � 180° � 360° 2 triangles
Find the sum of the angle measures in a pentagon.
Divide the figure into triangles.
3 � 180° � 540° 3 triangles
Look for a pattern between the number of sides and the number of triangles.
polygon
7-4 Classifying Polygons 341
7-4 Classifying Polygons
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The pattern is that the number of triangles is always 2 less than thenumber of sides. So an n-gon can be divided into n � 2 triangles. Thesum of the angle measures of any n-gon is 180°(n � 2).
All the sides and angles of a have equal measures.
Finding the Measure of Each Angle in a Regular Polygon
Find the angle measures in each regular polygon.
5 congruent angles 6 congruent angles
5x° � 180°(5 � 2) 6y ° � 180°(6 � 2)
5x° � 180°(3) 6y ° � 180°(4)
5x° � 540° 6y ° � 720°
�55x°� � �
5450°� �
66y°� � �
7260°�
x° � 108° y ° � 120°
Quadrilaterals with certain properties are given additional names. Ahas exactly 1 pair of parallel sides. A has 2 pairs
of parallel sides. A has 4 right angles. A has 4congruent sides. A has 4 congruent sides and 4 right angles.square
rhombusrectangleparallelogramtrapezoid
regular polygon
Quadrilaterals
Parallelograms2 pairs of parallel sides
Trapezoidsexactly 1 pair of
parallel sides
Rhombuses4 congruent sides
Rectangles4 right angles
Squares4 congruent sides
4 right angles
x°
x° x°
x° x°
y° y°
y° y°
y° y°
2E X A M P L E
342 Chapter 7 Foundations of Geometry
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Find the sum of the angle measures in each figure.
1. 2. 3.
Find the angle measures in each regular polygon.
4. 5. 6.
KEYWORD: MT7 Parent
KEYWORD: MT7 7-4
GUIDED PRACTICE
See Example 2
E F
H G
EF || GH
W X
Z Y
Think and Discuss
1. Choose which is larger, an angle in a regular heptagon or an anglein a regular octagon. Justify your answer.
2. Explain why all rectangles are parallelograms and why all squaresare rectangles.
Classifying Quadrilaterals
Give all of the names that apply to each figure.
quadrilateral Four-sided polygon
trapezoid 1 pair of parallel sides
quadrilateral Four-sided polygon
parallelogram 2 pairs of parallel sides
rectangle 4 right angles
E X A M P L E 3
t° t°
t° t°
v° v°
v°v°
v°v° v°
b° b°
b° b°
b° b°
See Example 1
7-4 Classifying Polygons 343
7-4 ExercisesExercises
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Give all of the names that apply to each figure.
7. 8. 9.
Find the sum of the angle measures in each figure.
10. 11. 12.
Find the angle measures in each regular polygon.
13. 14. 15.
Give all of the names that apply to each figure.
16. 17. 18.
Find the sum of the angle measures in each regular polygon. Then find themeasure of each angle.
Sketch a quadrilateral to fit each description. If no quadrilateral can be drawn, write not possible.
36. a parallelogram that is not a rhombus
37. a square that is not a rectangle
38. Earth Science Precious stones are oftencut in a brilliant cut to maximize the lightthey reflect. The best angles for a cutdepend on the type of stone. The bestangles for a diamond are shown.
a. If the pavilion main angle is 41°, find x.
b. If the crown angle is 35°, find y.
39. Architecture Fernando is designing a house. He wants one room to be in theshape of an irregular heptagon with two corners that form right angles. Whatangle measures could the remaining five corners have?
40. What’s the Error? A student said that all squares are rectangles, but not all squares are rhombuses. What was the error?
41. Write About It Why is it possible to find the sum of the angle measures of an n-gon using the formula (180n � 360)°?
42. Challenge Use a diagram and the properties of parallel lines to explainwhich angles in a parallelogram must be congruent.
43. Multiple Choice What is the measure of each angle of a regular 15-sided polygon?
146° 148° 150° 156°
44. Short Response The sum of the angle measures of a regular polygon is 720°. Name the regular polygon. What is the measure of each angle?
49. The first angle in a triangle is less than 90°. The second angle is �34� as large as
the first angle. The third angle is �23� as large as the second angle. Find the angle
measures and draw a possible figure. (Lesson 7-3)
DCBA
y ˚y ˚
x ˚
35˚
41˚ 41˚
35˚Crown angle
Pavilion mainangle
Earth Science
The ImperialState Crown ofGreat Britaincontains over3000 preciousstones, including2800 diamonds.
7-4 Classifying Polygons 345
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Exterior Angles of a Polygon
Use with Lesson 7-4
7-4
The exterior angles of a polygon are formed by extending the polygon’s sides. Every exterior angle is supplementary to the angle next to it inside the polygon.
Follow the steps to find the sum of the exterior angle measures for a polygon.
a. Use geometry software to make a pentagon. b. Use the LINE-RAY tool to extend Label the vertices A through E. the sides of the pentagon. Add
points F through J as shown.
c. Use the ANGLE MEASURE tool to measure d. Drag vertices A through E and watch the each exterior angle and the CALCULATOR sum. Notice that the sum of the angle tool to add the measures. Notice the sum. measures is always 360°.
1. Suppose you were to drag the vertices of a polygon so that the polygon almost vanishes. How would this show that the sum of the exterior angle measures is 360°.
1. Use geometry software to draw any polygon. Find the sum of its exteriorangle measures. Drag its vertices to check that the sum is always the same.
Activity
KEYWORD: MT7 Lab7
Exterior angle
Think and Discuss
Try This
1
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Learn to identifypolygons in thecoordinate plane.
Vocabulary
run
rise
slope
C
D
–3
6
Negative slope
x
y
L
MK N
x
y
In computer graphics, a coordinate system is used to create images, from simple geometric figures to realistic figures used in movies.
Properties of the coordinate plane can be used to find information about figures in the plane, such as whether lines in the plane are parallel.
is a number that describes how steep a line is.
slope � � �r
r
i
u
s
n
e�
slope of A�B� � �86� � �
43�
slope of C�D� � ��63� � �
�21�
The slope of a horizontal line is 0. The slope of a vertical line is undefined.
Finding the Slope of a Line
Determine if the slope of each line is positive, negative, 0, or undefined. Then find the slope of each line.
KL���
positive; slope of KL��� � �21� � 2
LM���
undefined; slope of LM��� � �10�
LN���
negative; slope of LN��� � ��42� � ��
12�
KM���
0; slope of KM��� � �01�
vertical change���horizontal change
Slope
When a number is divided by zero,the quotient isundefined. There is no answer.
7-5 Coordinate Geometry 347
7-5 Coordinate Geometry
A
B
Positive slope
x
y
E X A M P L E 1
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Finding Perpendicular and Parallel Lines
Which lines are parallel? Which lines are perpendicular?
slope of PQ��� � �32�
slope of RS��� � �43�
slope of AB��� � �32�
slope of PA��� � ��22� or �1
slope of GH��� � ��43�
slope of XY���� � ��87�
PQ���⏐⏐AB��� The slopes are equal: �32� � �
32�
RS��� ⊥ GH��� The slopes have a product of �1: �43� � �
�43� � �1
Using Coordinates to Classify Quadrilaterals
Graph the quadrilaterals with the given vertices. Give all of thenames that apply to each quadrilateral.
J�K�⏐⏐M�L� and M�J�⏐⏐L�K� S�P�⏐⏐R�Q�J�K� ⊥ L�K�, J�K� ⊥ M�J�, trapezoid
M�L� ⊥ L�K� and M�L� ⊥ M�J�parallelogram, rectangle, square, rhombus
A
x
y
X
S
Q
BP
H
G
Y
R
y
Undefinedslope
Undefinedslope
Slope = 0
Slope = 0M
KJ
L
x x
y
P
S
R
Slope = – 13
Slope = 1
Slope = 1
Slope = –1
Q
If a line has slope
�ba
�, then a line
perpendicular to it
has slope ��ba�.
Slopes of Parallel and Perpendicular Lines
Two lines with equal slopes are parallel.
Two lines whose slopes have a product of �1 are perpendicular.
2E X A M P L E
E X A M P L E 3
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7-5 ExercisesExercises
Determine if the slope of each line is positive,negative, 0, or undefined. Then find the slope ofeach line.
1. AD��� 2. BE���
3. MN��� 4. EF���
5. Which lines are parallel?
6. Which lines are perpendicular?
Graph the quadrilaterals with the given vertices. Giveall of the names that apply to each quadrilateral.
7. D(�3, �2), E(�3, 3), F(2, 3), G(2, �2)
8. R(�4, �1), S(�2, 2), T(4, 2), V(5, �1)
Find the coordinates of the missing vertex.
9. rhombus ABCD with A(2, 3), B(3, 1), and D(1, 1)
10. square JKLM with J(�3, 1), K(0, 1), and L(0, �2)
KEYWORD: MT7 Parent
KEYWORD: MT7 7-5
GUIDED PRACTICE
See Example 2
See Example 3
See Example 1
Finding the Coordinates of a Missing Vertex
Find the coordinates of the missing vertex of square ABCD.Square ABCD with A(4, 0), B(0, 4),and C(�4, 0)
Step 1 Graph and connect thegiven points.
Step 2 Complete the figure to find the missing vertex.A�B� has a slope of �1, so C�D� has a slope of �1. B�C�has a slope of 1, so A�D�has a slope of 1.
The coordinates of D are (0, �4).
C A
D
B
O 4
4
2
2�2
�2
�4
�4
x
y
Think and Discuss
1. Explain how you can use slopes to classify a quadrilateral.
4E X A M P L E
x
y
D
M
N
A
F B
E
C
In a square oppositesides are parallel.
7-5 Coordinate Geometry 349
See Example 4
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Determine if the slope of each line is positive,negative, 0, or undefined. Then find the slope of each line.
11. AB��� 12. EG���
13. HG��� 14. CH���
15. Which lines are parallel?
16. Which lines are perpendicular?
Graph the quadrilaterals with the given vertices.Give all of the names that apply to each quadrilateral.
17. D(�4, 3), E(4, 3), F(4, �5), G(�4, �5)
18. W(�2, 1), X(�2, �2), Y(4, 1), Z(0, 2)
Find the coordinates of the missing vertex.
19. rectangle ABCD with A(�3, 3), B(4, 3), and D(�3, �1)
20. trapezoid JKLM with J(�1, 5), K(2, 3), and L(2, 1)
Draw the line through the given points and find its slope.
21. A(1, 0), B(2, 3) 22. C(�3, 0), D(�3, �4)
23. G(4, �2), H(�1, �2) 24. E(�2, 1), F(3, �2)
25. A line passes through the coordinates P(1, 3) and Q(�2, �3). Identify the slope of PQ���. Then name two coordinates and the slope of a line perpendicular to PQ���.
26. AB���⏐⏐CD��� and the slope of AB��� is undefined. What can you tell about the slope of CD���? Explain.
27. On a coordinate grid draw a line s with slope 0 and a line t with slope 1. Then draw three lines through the intersection of lines s and t that have slopes between 0 and 1.
28. On a coordinate grid draw a line m with slope 0 and a line n with slope �1.Then draw three lines through the intersection of lines m and n that haveslopes between 0 and �1.
29. Critical Thinking Square ABCD has vertices at (1, 2) and (1, �2). Findthe possible coordinates of the two missing vertices to create the squarewith the least area. Justify your solution.
30. Critical Thinking Triangle LMN has vertices at L(�2, 2), M(0, 0), andN(�5, �1). What kind of triangle is it? Explain.
PRACTICE AND PROBLEM SOLVING
INDEPENDENT PRACTICE
Extra PracticeSee page 794.
See Example 2
See Example 3
See Example 4
See Example 1
x
y
H
D
C
B
E
A
GF
350 Chapter 7 Foundations of Geometry
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Tell if each statement is true or false. If it is false, give a counterexample.
31. Opposite sides of a rhombus have the same slope.
32. All of the adjacent sides of quadrilaterals have slopes with a product of �1.
33. All parallelograms have two pairs of lines with the same slope and adjacent sides that have slopes with a product of �1.
34. A trapezoid has two pairs of sides that have the same slope.
35. The slope of a horizontal line is always 0.
36. The slope of a line through the origin is always defined.
Identify and name each figure.
37. This figure has two sides with undefined slopes.
38. This figure has a side with a slope of �1.
39. This figure has a side with a slope of 3.
40. This figure has a side with a slope of �13�.
41. What’s the Question? Points P(3, 7),Q(5, 2), R(3, �3), and S(1, 2) form thevertices of a polygon. The answer is thatthe segments are not perpendicular. What is the question?
42. Write About It Explain how using different points on a line to find theslope affects the answer.
43. Challenge Use a square in a coordinate plane to explain why a line withslope 1 makes a 45° angle with the x-axis.
44. Multiple Choice A right triangle has vertices at (0, 0), (0,4), and (10, 4). What is the slope of the hypotenuse?
2.5 2 1.8 0.4
45. Gridded Response Find the slope of the line that crosses through the points A(2, 4) and B(�1, 5).
Find each number. (Lesson 6-4)
46. 60% of what number is 12? 47. 112 is 80% of what number?
48. 30 is 2% of what number? 49. 90% of what number is 18?
Find the sum of the angle measures of each polygon. (Lesson 7-4)
50. 15-gon 51. hexagon 52. n-gon 53. decagon
DCBA
x
y
4 5321
345
21
�3�4�5 �2�1
�2�1
�3�4�5
O
A
B
D
C
7-5 Coordinate Geometry 351
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Quiz for Lessons 7-1 Through 7-5
7-1 Points, Lines, Planes, and Angles
Use the diagram to name each figure.
1. two pairs of complementary angles
2. three pairs of supplementary angles
3. two right angles
7-2 Parallel and Perpendicular Lines
In the figure, line m⏐⏐line n. Find the measure of each angle.
4. �1 5. �2 6. �3
7-3 Angles in Triangles
Find x° in each triangle.
7. 8.
9. In �ABC, m�A � 57°, and �B is a right angle. What is m�C?
7-4 Classifying Polygons
Give all of the names that apply to each figure.
10. 11.
7-5 Coordinate Geometry
Graph the quadrilaterals with the given vertices. Give all of the names that apply to each quadrilateral.
14. square ABCD with A(�1, 1), B(2, 1), and C(2, �2)
15. parallelogram PQRS with P(3, 3), Q(4, 2), and R(2, �2)
Rea
dy
to G
o O
n?
352 Chapter 7 Foundations of Geometry
CB
D
A
E
F
35°
15°
75° 55°
m
n
125°
3 4
1 2
83°
x°60°
74°
x° x°
3 in.3 in.
3 in.
P
N
OM
3 in.A B
D C
AB || CD
m807_c07_352 1/13/06 11:55 AM Page 352
In the figure, �1 and �2 are complementary, and �1 and �5 are supplementary. If m�1 � 60°, find m�3 � m�4.
In triangle ABC, m�A � 35° and m�B � 55°. Use the Triangle Sum Theorem to determine whether triangle ABC is a right triangle.
The second angle in a quadrilateral is eighttimes as large as the first angle. The thirdangle is half as large as the second. Thefourth angle is as large as the first angle andthe second angle combined. Find the anglemeasures in the quadrilateral.
Parallel lines m and n are intersected by atransversal, line p. The acute angles formed by line m and line p measure 45°.Find the measure of the obtuse anglesformed by the intersection of line n and line p.
Write each problem in your own words. Check to make sure youhave included all of the information needed to solve the problem.
Understand the Problem• Restate the problem in your own words
If you write a problem in your own words, you may understand itbetter. Before writing a problem in your own words, you may needto read it over several times—perhaps aloud, so you can hearyourself say the words.
Once you have written the problem in your own words, you maywant to make sure you included all of the necessary information tosolve the problem.
Focus on Problem Solving 353
m
n
p
34
51 2
55°35°A
C
B
2
3
4
1
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Learn to use propertiesof congruent figures tosolve problems.
Vocabulary
correspondence
Below are the DNA profilesof two pairs of twins. TwinsA and B are identical twins.Twins C and D are fraternaltwins.
A is a way of matching up two sets of objects. The bands of DNA that are next to each other in each pair match up, or correspond. In the DNA of the identical twins, the corresponding bands are the same.
If two polygons are congruent, all of their corresponding sides and angles are congruent.
Writing Congruence Statements
Write a congruence statement for each pair of congruent polygons.
In a congruence statement, the vertices in the second triangle haveto be written in order of correspondence with the first triangle.
�K corresponds to �R. �K � �R�L corresponds to �Q. �L � �Q�M corresponds to �S. �M � �S
The congruence statement is triangle KLM � triangle RQS.
correspondence
A
B
C
D
65°
94°
21°
K
M L
65°
94°21°
Q S
R
E X A M P L E 1
Marks on the sides ofa figure can be usedto show congruence.K�M� � R�S� (1 mark)K�L� � R�Q� (2 marks)M�L� � S�Q� (3 marks)
Learn to transformplane figures usingtranslations, rotations,and reflections.
Vocabulary
image
reflection
center of rotation
rotation
translation
transformation
When you are on an amusementpark ride, you are undergoing atransformation. A is a change in a figure’s position orsize. Ferris wheels and merry-go-rounds are rotations. Free-fall ridesand water slides are translations.Translations, rotations, andreflections are types oftransformations.
The resulting figure, or , of a translation, rotation, or reflectionis congruent to the original figure.
Identifying Transformations
Identify each as a translation, rotation, reflection, or none of these.
translation none of these
\
rotation reflection
image
transformation
Translation Rotation Reflection
A slides a A turns a A flips a figure along a line figure around a point, figure across a line to without turning. called the create a mirror image.
.rotationcenter of
reflectionrotationtranslation
A� is read “A prime.”The point A� is theimage of point A.
E X A M P L E 1
P N
MKL
A BCF
DE
A′ B′C′
D′E′F′
J
H
G
E
FG′
H′
J′E′
F′
Q T
RS
P′
L′
K′
M′ N′
T′
S′R′
Q′
358 Chapter 7 Foundations of Geometry
7-7 Transformations
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Graphing Transformations
Draw the image of a triangle with vertices A(1, 1), B(1, 4), and C(3, 4) after each transformation.
translation 5 units down reflection across the y-axis
Describing Graphs of Transformations
Parallelogram EFGH has vertices E(�2, 1), F(3, 1), G(4, 4), and H(�1, 4). Find the coordinates of the image of the indicated pointafter each transformation.
translation 2 units down, 180° rotation around (0, 0),point E point G
E� (�2, �1) G� (�4, �4)
Think and Discuss
1. Tell whether the image of a vertical line is sometimes, always, ornever vertical after a translation, a reflection, or a rotation.
2. Describe what happens to the x-coordinate and the y-coordinateafter a point is reflected across the x-axis.
2E X A M P L E
E X A M P L E 3
x
y
�2�4
�2
�4
2
4
2
A
B C
A′
B′ C′
x
y
�2�4
�2
�4
2
4
2 4
A
B C
A′
C′ B′
H G
E
O 42
�2
�4
�4
x
y
FH′
E′
G′
F′
H G
E
O
2
2�2
�2
�4
�4
x
y
F
E′F′
G′ H′
7-7 Transformations 359
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7-7 ExercisesExercises
Identify each as a translation, rotation, reflection, or none of these.
1. 2.
Draw the image of the parallelogram ABCD with vertices (�3, 0), (�4, 3), (1, 4),and (2, 1) after each transformation.
3. translation 1 unit up 4. reflection across the x-axis
5. reflection across the y-axis 6. 180° rotation around (0, 0)
Triangle ABC has vertices A(2, 1), B(3, 3), and C(1, 2). Find the coordinatesof the image of the indicated point after each transformation.
7. translation 4 units down, point C 8. reflection across the x-axis, point B
9. reflection across the y-axis, point C 10. 180° rotation around (0,0), point A
Identify each as a translation, rotation, reflection, or none of these.
11. 12.
Draw the image of the quadrilateral ABCD with vertices (1, 1), (2, 4), (4, 5),and (5, 3) after each transformation.
13. translation 5 units down 14. reflection across the x-axis
15. reflection across the y-axis 16. 180° rotation around (0, 0)
Square ABCD has vertices A(�2, 2), B(2, 2), C(2, �2), and D(�2, �2). Find thecoordinates of the image of the indicated point after each transformation.
17. translation 3 units to the left, point A
18. translation 4 units to the right, point B
19. reflection across the x-axis, point C
20. 180° rotation around (0, 0), point A
KEYWORD: MT7 Parent
KEYWORD: MT7 7-7
GUIDED PRACTICE
INDEPENDENT PRACTICE
See Example 2
See Example 2
See Example 3
See Example 3
See Example 1
See Example 1
Q
U T
R
S
T′ U′
S′
R′Q′
J
L
M
K
M′K′
J′
L′
BC
A
C′
A′
B′
ZX
Y
X′
Y′
Z′
360 Chapter 7 Foundations of Geometry
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36. Multiple Choice Which best represents the transformation at right?
Translation
Rotation
Reflection
None of these
37. Short Response Draw the image of a triangle with vertices (�1, 2), (3, 3), and (1, �3) after a translation 2 units up and 2 units to the right.
Find each percent increase or decrease to the nearest percent. (Lesson 6-5)
38. from 75 to 90 39. from 1200 to 1400 40. from 44 to 21
Draw the line through the given points and find its slope. (Lesson 7-5)
Give the coordinates of each point after a 180° rotation around (0, 0).
30. (1, 2) 31. (�4, 5) 32. (m, n)
33. Write a Problem Write a problem involving transformations on acoordinate grid that result in a pattern.
34. Write About It Explain how each type oftransformation performed on the arrow would affect the direction the arrow is pointing.
35. Challenge A triangle has vertices (2, 5), (3, 7), and (7, 5). After a reflectionand a translation, the coordinates of the image are (7, �2), (8, �4), and(12, �2). Describe the transformations.
PRACTICE AND PROBLEM SOLVINGExtra Practice
See page 795.
m
n
Fm
n
L m
n
D
O 4
4
2�2
x
y
7-7 Transformations 361
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Combine Transformations
Use with Lesson 7-7
7-7
KEY
Pattern blocks �
triangle rhombus trapezoid
You can use a coordinate plane when transforming a geometric figure.
Follow the steps below to transform a figure.
a. Place a red pattern block on a coordinateplane. Trace the block, and label the vertices.
b. Translate the figure 3 units down and 5 unitsright, and then reflect the resulting figureacross the x-axis. Draw the image and labelthe vertices.
c. Now place a green pattern block on the samecoordinate plane. Trace the block and labelthe vertices. Rotate the figure 180° around thepoint (0, 0), and then translate it 4 units upand 3 units right. Draw the image and labelthe vertices.
1. When you perform two or more transformations on a figure, does itmatter in which order the transformations are performed? Explain.
1. Place a blue pattern block on a coordinate plane. Trace the block, andlabel the vertices. Perform two different transformations on the figure.Draw the image and label the vertices. Trade with a classmate.Describe the transformations your classmate used.
Activity 1
KEYWORD: MT7 Lab7
1
O 4 6 8
4
6
8
2
2�2
�2
�4
�6
�8
x
y
10 12
Think and Discuss
Try This
362 Chapter 7 Foundations of Geometry
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Follow the steps below to transform a figure.
a. Place a rhombus on a coordinate plane. Trace the rhombus, and label the vertices.
b. Rotate the figure 90° clockwise about the origin.
c. Reflect the resulting figure across the x-axis. Draw the image and label the vertices.
d. Now place a rhombus in the same position as the original figure. Reflect the figure across the line y � x.
1. What do you notice about the images that result from the twotransformations in parts b and c above and the image that resultsfrom the single transformation in part d above?
1. Place a pattern block on a coordinate plane. Trace the block and labelthe vertices. Perform two different transformations on the figure. Drawthe image and label the vertices. Explain what single transformation ofthe original figure would result in the same image.
Describe two different ways to transform each figure from position A to position B.
2. 3.
O 4
2
�2
�4
�4
x
A
y
BO 4
4
2
2�2
�2
�4
�4
x
A
y
B
Activity 2
1
Think and Discuss
Try This
O 4
4
2
2�2
�2
�4
�4
x
y
O 4
4
2
2�2
�2
�4
�4
x
y
7-7 Hands-On Lab 363
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Learn to identifysymmetry in figures.
Vocabulary
rotational symmetry
line of symmetry
line symmetry
E X A M P L E 1
If you fold a figureon the line ofsymmetry, the halvesmatch exactly.
Nature provides many beautiful examples ofsymmetry, such as the wings of a butterfly orthe petals of a flower. Symmetric objects haveparts that are congruent.
A figure has if you can draw aline through it so that the two sides are mirrorimages of each other. The line is called the
.
Drawing Figures with Line Symmetry
Complete each figure. The dashed line is the line of symmetry.
line of symmetry
line symmetry
7-8 Symmetry
364 Chapter 7 Foundations of Geometry
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A figure has if you can rotate the figure aroundsome point so that it coincides with itself. The point is the center ofrotation, and the amount of rotation must be less than one full turn,or 360°.
Complete each figure. The point is the center of rotation.
2-fold
Figure coincides with itself twice every full turn.
8-fold
Figure coincides with itself 8 times every full turn.
rotational symmetry
Think and Discuss
1. Explain what it means for a figure to be symmetric.
2. Tell which letters of the alphabet have line symmetry.
3. Tell which letters of the alphabet have rotational symmetry.
2E X A M P L E
7-fold and 6-foldrotational symmetrymean that the figurescoincide with themselves7 times and 6 timesrespectively, within onefull turn.
7-8 Symmetry 365
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7-8 ExercisesExercises
Complete each figure. The dashed line is the line of symmetry.
1. 2. 3. 4.
Complete each figure. The point is the center of rotation.
5. 4-fold 6. 6-fold 7. 3-fold
Complete each figure. The dashed line is the line of symmetry.
8. 9. 10.
11. 12. 13.
Complete each figure. The point is the center of rotation.
14. 4-fold 15. 5-fold 16. 2-fold
KEYWORD: MT7 Parent
KEYWORD: MT7 7-8
GUIDED PRACTICE
INDEPENDENT PRACTICE
See Example 2
See Example 1
See Example 1
See Example 2
366 Chapter 7 Foundations of Geometry
m807_c07_364_367 1/13/06 11:56 AM Page 366
Kage Asa no ha Maru ni shichiyo Nito Nami
27. Short Answer Draw a figure that has line symmetry and rotational symmetry.
28. Multiple Choice Which figure has 90° rotational symmetry?
regular pentagon regular hexagon
square regular heptagon
Find each unit rate. (Lesson 5-2)
29. 20 bananas for $4.40 30. 496 miles in 16 hours 31. 20 oz for $3.20
In the figure, ABCDE � PQRST. (Lesson 7-6)
32. Find j. 33. Find k.
34. Find m. 35. Find n.
DB
CA
Draw an example of a figure with each type of symmetry.
17. line and rotational symmetry 18. no symmetry
How many lines of symmetry do the following figures have?
19. square 20. rectangle
21. equilateral triangle 22. isosceles triangle
23. Social Studies Family crests called ka-mon have been in use in Japan formany centuries. Copy each crest below. Describe the symmetry, and drawany lines of symmetry or the center of rotation.
a. b. c.
24. Write a Problem Signal flags are hung from lines of rigging on ships.Write a problem about the types of symmetry in the flags.
25. Write About It To complete a figure with n-fold rotational symmetry,explain how much you rotate each part.
26. Challenge The flag of Switzerland has 180°rotational symmetry. Identify at least three other countries that have flags with 180°rotational symmetry.
PRACTICE AND PROBLEM SOLVINGExtra Practice
See page 795.
EB
D
A
C
4m � 3
2.5k
21
2j � 4T
Q
S
P
R
13
10
3n
14
7-8 Symmetry 367
Social Studies
In Japan, akimono thatdisplays thewearer’s familycrest is wornfor ceremonialoccasions.
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Learn to createtessellations.
Vocabulary
regular tessellation
tessellation
Fascinating designs can be made byrepeating a figure or group of figures.These designs are often used in art andarchitecture.
A repeating pattern of plane figures thatcompletely covers a plane with no gapsor overlaps is a .
In a , a regularpolygon is repeated to fill a plane. Theangle measures at each vertex must add to 360°, so only three regulartessellations exist.
Equilateral triangles Squares Regular hexagons
6 � 60° � 360° 4 � 90° � 360° 3 � 120° � 360°
It is also possible to tessellate with polygons that are not regular. Sincethe angle measures of a triangle add to 180°, six triangles meeting ateach vertex will tessellate. The angle measures of a quadrilateral add to360°, so four quadrilaterals meeting at a vertex will tessellate.
Creating a Tessellation
Create a tessellation withquadrilateral ABCD.
There must be a copy of each angleof quadrilateral ABCD at every vertex.
regular tessellation
tessellation
E X A M P L E 1B
A
D C
368 Chapter 7 Foundations of Geometry
7-9 Tessellations
Alcazar Palace in Seville, Spain
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Think and Discuss
1. Explain why a regular pentagon cannot be used to create a regulartessellation.
2. Describe the transformations used to make the tessellation inExample 2.
2E X A M P L E
1. Create a tessellation with quadrilateral QRST.
2. Use rotations to create a variation of the tessellation in Exercise 1.
KEYWORD: MT7 Parent
KEYWORD: MT7 7-9
GUIDED PRACTICE
See Example 2
See Example 1
Q
S
T
R
Creating a Tessellation by Transforming a Polygon
Use rotations to create a variation of the tessellation in Example 1.Step 1: Find the midpoint of a side.
Step 2: Make a new edge for half of the side.
Step 3: Rotate the new edge around the midpoint to form the edge of the other half of the side.
Step 4: Repeat with the other sides.
Step 5: Use the figure to make a tessellation.
BA
D C
BA
D C
BA
D C
7-9 Tessellations 369
7-9 ExercisesExercises
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Use each shape to create a tessellation.
5. 6. 7.
8. 9. 10.
11. A piece is removed from one side of a rectangle and translated to the opposite side. Will this shape tessellate?
12. A piece is removed from one side of a trapezoid and translated to the opposite side. Will this shape tessellate?
13. In a semiregular tessellation, two or more regular polygons are repeated to fill the plane and the vertices are all identical. Use each arrangement of regular polygons to create a semiregular tessellation.
a. b. c.
PRACTICE AND PROBLEM SOLVINGExtra Practice
See page 795.
3. Create a tessellation with triangle PQR.
4. Use rotations to create a variation of the tessellation in Exercise 3.
M. C. Escher created works of art by repeating interlocking shapes.He used both regular and nonregular tessellations. He often usedwhat he called metamorphoses, in which shapes change into othershapes. Escher used his reptile pattern in many hexagonaltessellations. One of the most famous is entitled simply Reptiles.
14. The steps below show the method Escher used to make abird out of a triangle. Use the bird to create a tessellation.
15. Critical Thinking What regular polygon do you think Escher used to begin Reptiles?
16. Challenge Create an Escher-like tessellation of your own design.
17. Multiple Choice Which of the following shapes will NOT form a regular tessellation?
18. Short Answer Which set of polygons will create a tessellation? Explain.
Write each number in scientific notation. (Lesson 4-4)
19. 3,400,000,000 20. 0.00000045 21. 28,000
Tell whether the two lines described in each exercise are parallel, perpendicular, orneither. (Lesson 7-5)
22. PQ��� has slope �32�. EF��� has slope ��
23�. 23. AB��� has slope �1
91�. CD��� has slope ��
34�.
DCBA
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Quiz for Lessons 7-6 Through 7-9
7-6 Congruence
In the figure, triangle ABC � triangle LMN.
1. Find q. 2. Find r. 3. Find s.
7-7 Transformations
Identify each as a translation, rotation, reflection, or none of these.
4. 5.
Quadrilateral ABCD has vertices A(�7, 5), B(�4, 5), C(�2, 3), and D(�6, 2).Find the coordinates of the image of each point after each transformation.
6. translation 4 units down, point C 7. reflection across the y-axis, point A
7-8 Symmetry
8. Complete the figure. The dashed line 9. Complete the figure with 4-fold is the line of symmetry. rotational symmetry. The point is
the center of rotation.
7-9 Tessellations
10. Copy the given figure and use it to create a tessellation.
Rea
dy
to G
o O
n?
372 Chapter 7 Foundations of Geometry
C N M
A
65°
25°
135q°
28 7s
r � 9
B
L
U
T
RQ
S
U′T′
Q′R′
S′
KN
L
J
M
N′
L′M′
J′
K′
m807_c07_372_372 1/13/06 11:57 AM Page 372
Mu
lti-Step Test Prep
Multi-Step Test Prep 373
Cloth Creations The Asante people of Ghana are knownfor weaving Kente cloth, a colorful textile based on repeatinggeometric patterns. Susan is using a coordinate plane todesign her own Kente cloth pattern.
1. Susan starts with triangle ABCas shown. Explain how she canuse slopes to make sure thetriangle is a right triangle.
2. To begin the pattern, Susan uses transformations to make a row of triangles that are allcongruent to triangle ABC.Describe the transformations she should use to make these triangles.
3. Next, she extends the pattern by making additionalrows of triangles. The firsttriangle in Row 2 is shown.Complete the table by writingthe coordinates of the topvertex of each triangle in the pattern.
4. What patterns do you notice in the table?
5. Susan’s Kente cloth pattern is a tessellation of what types of figures?
x
y
22242628210
2
2 4 6O
C
A B
x
y
22242628210
2
2 4 6O
C
A B
x
y
22242628210
4
6
2
2 4 6O
Row 4
Row 3
Row 2
Row 1
C
A B
Row Top Vertex of Triangles in Row
Row 1 (�9, 2) (�4, 2) (1, 2) (6, 2)
Row 2 (�8, 4)
Row 3
Row 4
m807_c07_373 1/13/06 11:57 AM Page 373
The object of this game is to creategeometric figures. Each card in the deckshows a property of a geometric figure. Tocreate a figure, you must draw a polygonthat matches at least three cards in yourhand. For example, if you have the cards“quadrilateral,” “a pair of parallel sides,” and“a right angle,” you could draw a rectangle.
A complete set of rules and playing cards is available online.
Coloring TessellationsTwo of the three regular tessellations—triangles and squares—can becolored with two colors so that no two polygons that share an edgeare the same color. The third—hexagons—requires three colors.
1. Determine if each semiregular tessellation can be colored withtwo colors. If not, tell the minimum number of colors needed.
2. Try to write a rule about which tessellations can be coloredwith two colors.
Polygon RummyPolygon Rummy
KEYWORD: MT7 Games
374 Chapter 7 Foundations of Geometry
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A
B
Project CDGeometry
PROJECT
Materials • 3 sheets of white
paper• CD or CD-ROM• scissors• tape• markers• empty CD case
It’s in the Bag! 375
C
Make your own CD to record important factsabout plane geometry.
Fold a sheet of paper in half. Place a CD on topof the paper so that it touches the folded edge.Trace around the CD. Figure A
Cut out the CD shape, being careful to leave thefolded edge attached. This will create two paperCDs that are joined together. Cut a hole in thecenter of each paper CD. Figure B
Repeat steps 1 and 2 with the other two sheetsof paper.
Tape the ends of the paper CDs together tomake a string of six CDs. Figure C
Accordion fold the CDs to make a booklet.Write the number and name of the chapter onthe top CD. Store the CD booklet in an emptyCD case.
Taking Note of the MathUse the blank pages in the CD booklet to take notes on the chapter. Be sure to include definitions and sample problems that will help you review essential concepts about plane geometry.
5
4
3
2
1
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376 Chapter 7 Foundations of Geometry
Complete the sentences below with vocabulary words from thelist above.
1. Lines in the same plane that never meet are called ___?___.Lines that intersect at 90° angles are called ___?___.
2. A quadrilateral with 4 congruent angles is called a ___?___. A quadrilateral with 4 congruent sides is called a ___?___.
Vocabulary
Points, Lines, Planes, and Angles (pp. 324–328)7-1
E X A M P L E EXERCISES
■ Find the angle measure.
m�1m�1 � 122° � 180°
� 122° � 122°���� ��m�1 � 58°
Find each angle measure.
3. m�1
4. m�2
5. m�3
Stu
dy
Gu
ide:
Rev
iew
. . . . . . . . . . . . 325
. . . . . . . . . . 336
. . . . . . . . . . . . . . . . . . 325
. . . . . . 358
. . 325
. . . . . . . . . . . . . 325
. . . . . . . . 354
. . . . . 337
. . . . . . . . . . . . . . . . . 358
. . . . . . . 337
. . . . . . . . . . . . . . . . . . . 324
. . . . . . . 364
. . . . . . . . . 364
. . . . . . . . . . . 325
. . . . . . . . . 336
. . . . . . . . . . . 330
. . . . . . . . . . 342
. . . . 330
. . . . . . . . . . . . . . . . . . 324
. . . . . . . . . . . . . . . . . . 324
. . . . . . . . . . . . . . . 341
. . . . . . . . . . . . . . . . . . . . 324
. . . . . . . . . . . . . . 342
. . . . . . . . . . . . . . 358
. . . . . . . . 342
. . . . . 368
. . . . . . . . . . . . . . 342
. . . . . . . . . . . . . 325
. . . . . . . . . . 336
. . . . . . . . . . . . . . . . . . . 347
. . . . . . . . . . . . . . . 358
. . . 365
. . . . . . . . . . . . . . . . . . . 347
. . . . . . . . 337
. . . . . . . . . . . . . . . 324
. . . . . . . . . . . . . . . . . . 347
. . . . . . . . . . . . . . . . 342
. . 325
. . . . . . . . . . . . 368
. . . . . . . . 358
. . . . . . . . . . . . 358
. . . . . . . . . . . . 330
. . . . . . . . . . . . . . 342
. 336
. . . . . . . . . 325vertical angles
Triangle Sum Theorem
trapezoid
transversal
translation
transformation
tessellation
supplementary angles
square
slope
segment
scalene triangle
run
rotational symmetry
rotation
rise
right triangle
right angle
rhombus
regular tessellation
regular polygon
reflection
rectangle
ray
polygon
point
plane
perpendicular lines
parallelogram
parallel lines
obtuse triangle
obtuse angle
line symmetry
line of symmetry
line
isosceles triangle
image
equilateral triangle
correspondence
congruent
complementary angles
center of rotation
angle
acute triangle
acute angle
1 122°2 3 1
23
68°
m807_c07_376_378 1/13/06 11:57 AM Page 376
Parallel and Perpendicular Lines (pp. 330–333)7-2
E X A M P L E EXERCISES
Line j⏐⏐line k. Find each angle measure.
■ m�1 m�1 � 143°
■ m�2 m�2 � 143° � 180°
� 143° � 143°���� ��m�2 � 37°
Line p⏐⏐line q. Find each angle measure.
6. m�1
7. m�2
8. m�3
9. m�4
10. m�5
Coordinate Geometry (pp. 347–351)7-5
E X A M P L E EXERCISES
■ Graph the quadrilateral with the givenvertices. Give all the names that apply.D(�2, 1), E(2, 3), F(3, 1), G(�1, �1)
D�E�⏐⏐F�G�E�F� ⏐⏐G�D�D�E� ⊥ E�F�
Graph the quadrilaterals with the givenvertices. Give all the names that apply.
14. Q(2, 0), R(�1, 1), S(3, 3), T(8, 3)
15. K(2, 3), L(3, 0), M(2, �3), N(1, 0)
16. W(2, 2), X(2, �2), Y(�1, �3), Z(�1, 1)
Stud
y Gu
ide: R
eview
143° 21
j
k
50°
n°
x
y
D
G
E
Fparallelogram,rectangle
1
p
q
2 3
4 5114°
128°
m°
m°
Angles in Triangles (pp. 336–340)7-3
E X A M P L E EXERCISES
■ Find n°.
n° � 50° � 90° � 180°n° � 140° � 180°
� 140° � 140°����� ��
n° � 40°
11. Find m°.
Classifying Polygons (pp. 341–345)7-4
E X A M P L E EXERCISES
■ Find the angle measures in a regular 12-gon.
12x° � 180°(12 � 2)12x° � 180°(10) 12x° � 1800°
x° � 150°
Find the angle measures in each regularpolygon.
12. a regular octagon
13. a regular 11-gon
Study Guide: Review 377
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Stu
dy
Gu
ide:
Rev
iew
Congruence (pp. 354–357)7-6
E X A M P L E EXERCISES
■ Triangle ABC � triangle FDE. Find x.
x � 4 � 4� 4 � 4�� ��
x � 8
Triangle JQZ � triangle VTZ.
17. Find x.
18. Find t.
19. Find q.
Transformations (pp. 358–361)7-7
E X A M P L E EXERCISES
■ Draw the image of a triangle with vertices (�2, 2),(1, 1), and (�3, �2) after a 180°rotation around (0, 0).
Draw the image of a triangle ABC withvertices (1, 1), (1, 4), and (3, 1) after eachtransformation.
20. reflection across the x-axis
21. translation 5 units left
22. 180° rotation around (0, 0)
Symmetry (pp. 364–367)7-8
E X A M P L E EXERCISES
■ Complete the figure. The dashed line isthe line of symmetry.
Complete each figure.
23. 6-fold 24. 25.
Tessellations (pp. 368–371)7-9
E X A M P L E EXERCISES
■ Create a tessellation with the figure.
Create a tessellation with each figure.
26. 27.
35
4
D
E F
x � 4A
B
C
x
y
48°7.2
(3x)°
3t
VZ
Q T
J 75°2q � 1 13
378 Chapter 7 Foundations of Geometry
m807_c07_376_378 1/13/06 11:57 AM Page 378
Chapter 7 Test 379
In the figure, line m⏐⏐line n.
1. Name two pairs of supplementary angles.
2. Find the m�1.
3. Find the m�2.
4. Find the m�3.
5. Find the m�4.
6. Two angles in a triangle have measures of 44° and 57°. What is the measureof the third angle?
7. What are the measures of the congruent angles in an isosceles triangle if the measure of the third angle is 102°?
Give all of the names that apply to each figure.
8. 9.
Graph the quadrilateral with the given vertices. Give all of the names thatapply to each quadrilateral.
In the figure, quadrilateral ABCD � quadrilateral LMNO.
13. Find m.
14. Find n.
15. Find p.
Pentagon ABCDE has vertices A(1, �2), B(3, �1), C(7, �2), D(6, �4), and E(2, �5).Find the coordinates of the image of each point after each transformation.
16. rotation 90° around the origin, point E 17. reflection across the x-axis, point C
18. translation 6 units up, point B 19. reflection across the y-axis, point A
20. Complete the figure. The dashed line is the line of symmetry.
A
B
C
DAB || CDAD || BC
3 cm
3 cm
3 cm 3 cm Ch
apter Test
135°1 2
3 4
m
n
CD
131°
27
11
21
74°
n � 6
(m � 45)°
3p
AB
L M
O
N
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Test
Tac
kler
380 Chapter 7 Foundations of Geometry
Extended Response Julianna bought a shirt marked down 20%.She had a coupon for an additional 20% off the sale price. Is thisthe same as getting 40% off the regular price? Explain yourreasoning.
4-point response:
The student answers the question correctly and shows all work.
3-point response:
The student makes a minor computation error that results in an incorrect answer.
2-point response:
The student makes major computation errors and does not show all work.
1-point response:
The student shows no work and has the wrong answer.
It is the same.
No, it is not the same. A $30 shirt with 20% off and then an additional
20% off is $6. A $30 shirt at 40% off is $12.
Yes, it is the same. If the shirt originally cost $25, it would cost $15
after taking 20% off of a 20% discount. A 40% discount off $20 is $15.
Shirt original price � $25
Shirt at 20% off � $20 $25 � 20% � $5; $25 � $5 � $20
Shirt at 20% off sales price � $15 $20 � 20% � $4; $20 � $4 � $15
Shirt at 40% off � $15 $25 � 40% � $10; $25 � $10 � $15
No, the prices are not the same. Suppose the shirt originally cost $40.
20% off a 20% markdown: $40 � 20% � $8; $40 � $8 � $32;
$32 � 20% � $6.40; $32 � $6.40 � $25.60
40% off: $40 � 40% � $16; $40 � $16 � $24
Scoring Rubric
4 points: The studentanswers all parts of thequestion correctly,shows all work, andprovides a complete and correct explanation.
3 points: The studentanswers all parts of thequestion, shows allwork, and provides acomplete explanationthat demonstratesunderstanding, but thestudent makes minorerrors in computation.
2 points: The studentdoes not answer allparts of the questionbut shows all work andprovides a complete andcorrect explanation forthe parts answered, orthe student correctlyanswers all parts of thequestion but does notshow all work or doesnot provide anexplanation.
1 point: The studentgives incorrect answersand shows little or nowork or explanation, orthe student does notfollow directions.
0 points: The studentgives no response.
Extended Response: Write Extended ResponsesExtended response test items often consist of multi-step problems to evaluate your understanding of a math concept. Extended response questions are scored using a 4-point scoring rubric.
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Test Tackler 381
To receive full credit, make sure allparts of the problem are answered. Besure to show all of your work and towrite a neat and clear explanation.
Read each test item and answer thequestions that follow.
Item AJanell has two job offers. Job A pays $500per week. Job B pays $200 per week plus15% commission on her sales. Sheexpects to make $7500 in sales permonth. Which job pays better? Explainyour reasoning.
1. A student wrote this response:
What score should the student’sresponse receive? Explain yourreasoning.
2. What additional information, if any,should the student’s response includein order to receive full credit?
3. Add to the response so that itreceives a score of 4-points.
4. How much would Janell have tomake in sales per month for job Aand job B to pay the same amount?
Item BA new MP3 player normally costs $97.99.This week, it is on sale for 15% off itsregular price. In addition to this, Jasminereceives an employee discount of 20%off the sale price. Excluding sales tax,what percent of the original price willJasmine pay for the MP3 player?
5. What information needs to beincluded in a response to receive full credit?
6. Write a response that would receivefull credit.
Item CThree houses were originally purchasedfor $125,000. After each year, the value ofeach house either increased or decreased.Which house had the least value after thethird year? What was the value of thathouse? Explain your reasoning.
7. A student wrote this response:
What score should the student’sresponse receive? Explain yourreasoning.
8. What additional information, if any,should the student’s response includein order to receive full credit?
Item DKara is trying to save $4500 to buy a usedcar. She has $3000 in an account thatearns a yearly simple interest of 5%. Willshe have enough money in her accountafter 3 years to buy a car? If not, howmuch more money will she need? Explainyour reasoning.
9. What information needs to beincluded in a response to receive full credit?
10. Write a response that would receivefull credit.
House A increased 3% over three years.
House B increased 1% over three years.
House C increased 3% over three years. So,
House B had the least value after the third
year. Its value increased 1% of $125,000, or
$1250, for a total value of $126,250.
Job A pays better.
Test Tackler
House Original Year 1 Year 2 Year 3Cost ($)
A 125,000 1% 1% 1%
B 125,000 4% �2% �1%
C 125,000 3% �2% 2%
Percent Change in Value
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Stan
dar
diz
ed T
est
Prep
KEYWORD: MT7 TestPrep
1. Which angle is a right angle?
�FED �GEH
�FEG �GED
2. A jeweler buys a diamond for $68 andresells it for $298. What is the percentincrease to the nearest percent?
3% 138%
33% 338%
3. A grocery store sells one dozen ears ofwhite corn for $2.40. What is the unitprice for one ear of corn?
0.05/ear of corn
$0.20/ear of corn
$1.30/ear of corn
$2.40/ear of corn
4. The people of Ireland drink the mostmilk in the world. All together, theydrink more than 602,000,000 quartseach year. What is this number writtenin scientific notation?
60.2 � 105
602 � 106
6.02 � 108
6.02 � 109
5. Cara is making a model of a car that is14 feet long. What other information is needed to find the length of themodel?
Car’s width Scale factor
Car’s speed Car’s height
6. For which equation is the point asolution to the equation?
y � 2x � 1 y � �x � 1
y � 2x � 2 y � �2x � 2
7. What is q in the acute triangle?
62 118
72 128
8. Which expression represents “twice thedifference of a number and 5”?
10. Marcus bought a shirt that was on salefor 20% off its regular price. If Marcuspaid $20 for the shirt, what what itsregular price?
$25 $16
$40 $30
Gridded Response
Use the following figure for items 11 and12. Line p is parallel to line q.
11. What is the measure of �4, in degrees?
12. What is the sum of the measures of �2and �6, in degrees?
13. Maryann bought a purse on sale for25% off. She paid $36 for the pursebefore tax. How much did the pursecost originally?
14. What is the value of the expression�2xy � y2, when x � �1 and y � 4?
15. A parallelogram has vertices at A(�2, 4), B(�1, �1), C(1, 0), and D(0, 5). What is the x-coordinate of Bafter the parallelogram is reflectedover the y-axis?
16. Guillermo invests $180 at a 4% simpleinterest rate for 6 months. How muchmoney will Guillermo earn in interest?Write your answer as a decimal to thenearest tenth.
Short Response
17. Triangle ABC, with vertices A(2, 3), B(4, �5), C(6, 8), is reflected across the x-axis to form triangle A�B�C�.
a. On a coordinate grid, draw andlabel triangle ABC and triangleA�B�C�.
b. Give the new coordinates fortriangle A�B�C�.
18. Complete the table to show thenumber of diagonals for the polygonswith the numbers of sides listed.
Extended Response
19. Four people are introduced to eachother at a party, and they all shakehands.
a. Explain in words how the diagramcan be used to determine thenumber of handshakes exchangedat the party.
b. How many handshakes areexchanged?
c. Suppose that six people wereintroduced to each other at a party.Draw a diagram similar to the oneshown that could be used todetermine the number ofhandshakes exchanged.
JG
HF
Stand
ardized
Test Prep
Use logic to eliminate answer choicesthat are incorrect. This will help you tomake an educated guess if you arehaving trouble with the question.