Download Answers: Chapter Test Circles - CK

Geometry Teacher’s Edition - Assessment
CK-12 Foundation
January 20, 2010
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Contents
1 Geometry TE - Assessment
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1.1
Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1.2
Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1.3
Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1.4
Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1.5
Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1.6
Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1.7
Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1.8
Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1.9
Chapter 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1.10 Chapter 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1.11 Chapter 11 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
1.12 Chapter 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
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Chapter 1
Geometry TE - Assessment
1.1
Chapter 1
Quiz: Points, Lines, and Planes
Name:________________________ Hour:______ Date:______________
1. Describe the following diagram as completely as possible.
2. Why are point, line, and plane considered undefined terms? What does being “undefined”
allow for these terms?
3. Draw a picture of two intersecting planes and darken the intersection. What type of
geometric figure results from this intersection?
−−→
←→
4. Draw a sketch of the following information: CD intersecting AB at point Q.
5. What is the best way to describe the path taken by air from Houston, Texas to Min-
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neapolis, Minnesota?
Answers: Points, Lines, and Planes
←→
←→
1. CB intersects HG at point I.
2. Sample: There is no precise way to define these words. Leaving them undefined allows
the words to be recognizable but use them in other definitions, such as collinear.
3. Sample diagram is shown below. The resulting figure is a line.
http://commons.wikimedia.org/wiki/File:Interseting_planes.svg
4. Sample is shown below
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5. Line segment
Quiz: Segments, Distances, Rays, and Angles
Name:________________________ Hour:______ Date:______________
1. Marcia took a road trip with some friends. When she left, her odometer read 65, 253 miles.
When she returned, her odometer read 69, 107 miles. How many miles did Marcia drive?
2. Graph A = (1, 2) and B = (−6, 2) on the graph below. Find AB.
3. Find the measure of ∠T U V below.
4. Suppose in the drawing below, m∠SAD = 52◦ and m∠SAT = 64◦ . What can you
conclude using the Angle Addition Property?
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5. Using the diagram below list all pairs of equal angles.
6. Match the each of the following terms with its correct definition.
a. Straight Angle
b. Right Angle
c. Acute Angle
d. Obtuse Angle
I. 90◦ < m∠A < 180◦
II. m∠A = 180◦
III. 0◦ < m∠A < 90◦
IV. m∠A = 90◦
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Answers: Segments, Distances, Rays, and Angles
1. 3, 854 miles
2. AB = 7 units
3. Approximately 122◦
4. m∠DAT = 116◦
5. m∠ZQY = m∠RQP ; m∠RQZ = m∠P QY
6. a − 2, b − 4, c − 3, d − 1
Quiz: Segments and Angles
Name:________________________ Hour:______ Date:______________
1. Assume AB = 10 mm and CD = 10 mm. What can you conclude about these two
segments?
2. The segment below is 12.5 cm. Find and label its midpoint, marking congruent segments.
3. Construct ∠ABC with measure 127◦ . Then draw its angle bisector, labeling congruent
angles.
4. Assume ∠V ∼
= ∠R. Find x and the measure of each angle.
Answers: Segments and Angles
1. You can conclude AB ∼
= CD
2. Answer shown below. Each segment has length 6.25 cm.
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3. Answer shown below. Bisector should divide main angle into 63.5◦ sections.
4. x = 9; m∠V = m∠R = 42◦
Quiz: Angle Pairs
Name:________________________ Hour:______ Date:______________
1. Describe the difference between a pair of complementary angles and a pair of supplementary angles.
2. True or False. Supplementary angles always form a linear pair.
3. Using the diagram below, list the following:
a. A pair of vertical angles
b. A linear pair
c. A pair of complementary angles
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4. Assume the following diagram. m∠T M K = 55◦ .
a. Find m∠T M H
b. Find m∠JM H
5. Using the above diagram, now suppose m∠T M K = 11z + 3 and m∠JM H = 2z + 30.
Find m∠T M K.
Answers: Angle Pairs
1. Sample: Supplementary angles have a sum of 180 degrees while complementary angles
have a sum of 90 degrees. Supplementary angles form a straight angle and complementary
angles form a right angle.
2. False, a counter example can be drawn with two non-adjacent angles with a sum of
180 degrees.
3. Samples:
a. ∠ECA & ∠BCD
b. ∠EAC & ∠ECB
c. ∠ECF & ∠F CB
4. m∠T M K = 125, m∠JM H = 55
5. 36◦
Quiz: Classifying Polygons
Name:________________________ Hour:______ Date:______________
1. Label the following triangle as completely as possible.
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2. Draw an acute isosceles triangle.
3. Can the following be classified as a polygon? Explain your answer.
4. Calculate BC
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Answers: Classifying Polygons
1. Obtuse triangle. It cannot be assumed an isosceles because there are no tic marks showing
equal lengths.
2. Sample shown below.
3. Yes, the figure can be classified as a polygon. It is the union of line segments meeting at
endpoints. It is a closed figure with no curved sides.
√
4. BC = 34 units
Test Basics of Geometry
Name:________________________ Hour:______ Date:______________
1. Brad lives 7 miles north from Kevin. Corey lives 8 miles west of Kevin.
a. Show possible locations of Brad, Kevin, and Corey on a coordinate grid.
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b. How far does Brad live from Corey?
2. Underline the undefined term(s) in the following definition of coplanar. Two or more
lines are coplanar if they share the same plane.
3. Suppose M is the midpoint of AB. What is AB if M B = 6 miles?
4. Aaron visited his friend, Crystal. She explained to him how her town, Shelbyville, was
set up.
a. Using the following information, draw a map of Shelbyville.
• Main Street is parallel to Second Avenue.
• Second Street is parallel to Washington Avenue.
• Jackson Street is perpendicular to Washington Avenue.
b. What must be true about this map?
Use the figure for questions 5 - 7.
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5. Name three collinear points.
6. Name four coplanar points.
←
→
7. What is the intersection of HI and plane J?
8. Why is it useful to have more than one way to name an angle?
9. Suppose ∠BJM = 3x+4 and M KD = 5x−10. ∠BJM and ∠M KD are complementary.
Solve for x and determine the measurement of each angle.
10. Explain, using one of the postulates you have learned, why surveyors use three-legged
stands for their instruments.
11. Determine the distance between (−4, 7) and (14, 7).
12. Suppose C is the midpoint of AB. Find AC, CB, and AB.
13. Highway mile markers give the mileage from one end of a state to its opposite end. In
Michigan, US127 runs from the Ohio/Michigan state line to Clare. If Jackson is located at
mile marker 43 and Alma is located at mile marker 118, how far apart are these two cites?
14. Find the measure of this angle and classify it as obtuse, acute, right, or straight.
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15. Draw and label a figure to fit the following information: two angles that are supplementary, adjacent, and equal measure.
16. Is the following figure considered a polygon? Explain your reasoning. If it is a polygon,
name it appropriately.
17. Estimate the angle formed by the hour and minute hands of a clock at 9:35.
Answers: Test Basics of Geometry
1. See below
2. Two or more lines are coplanar if they share the same plane.
3. AB = 12 miles
4. See below
5. E, G, F
6. E, G, F, J
7. G
8. Sample: The intersection formed by two lines or segments results in four angles, so by
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using all three points when naming an angle, you can be very specific regarding the angle
you are referencing
9. x = 12; m∠BJM = 40; m∠M KD = 50
10. Sample: The ______ Postulate states that three points determine a plane. The legs
of the tripod represent three points and the ground represents the plane. Adding a fourth
leg (point) often makes the chair or stand wobbly.
11. 18 units
12. x = 5; AC = 15, BC = 15, AB = 30
13. 75 miles
14. 95 degrees
15. See below
16. Sample Yes, the figure is a polygon because it is a closed figure made with the union of
line segments. It’s specific name is a nonconvex (concave) heptagon.
17. 60 degrees
Standardized Test Prep Basics of Geometry
Name:________________________ Hour:______ Date:______________
Use the diagram to answer questions 1 - 3.
1. How many segments are in the figure?
A. 1
B. 4
C. 6
D. 10
E. 14
−−→
2. Which ray is opposite to DE?
−−→
A. BE
−−→
B. CB
−→
C. AE
−−→
D. DE
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−−→
E. DB
−−→
What is another name for BD?
−−→
A. DE
−−→
B. BE
−−→
C. CB
−→
D. CA
E. none of the above
4. Which figure could be the intersection of two planes?
A. line
B. ray
C. point
D. segment
E. A & B only
5. N U has endpoints (−8, 1) and (−8, −8). Which other point must lie on the segment?
A. (16, 7)
B. (0, −7)
C. (8, 3)
D. (−8, 3)
E. none of the above
6. Points A, B, and C are collinear with A between B and C. Which of the following must
be true?
A. AB + BC = AB
B. BC − AB = AC
C. AB − BC = AB
D. A and B only
E. A and C only
7. Two angles are adjacent and supplementary. What is the measure of each?
A. 90
B. 180
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C. 45
D. 60
E. cannot be determined
8. A triangle has three equal sides. What can you determine about its angles?
A. The measures are 30◦
B. The measures are 60◦
C. cannot be determined
9. ∠K and ∠F are vertical angles. Their sum is 146◦ . What is m∠K?
A. 73
B. 36.5
C. 144
D. 62
E. cannot be determined
10. The measure of an angle is 22 less than the measure of its supplement. What is the
measure of the angle?
A. 101
B. 79
C. 148
D. 123
E. none of the above
11. Which term is a geometric undefined term?
A. Point
B. line
C. plane
D. A and B only
E. all of the above
12. Which best describes the figure below?
A. Scalene acute triangle
B. equilateral acute triangle
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C. Isosceles obtuse triangle
D. scalene obtuse triangle
E. isosceles acute triangle
Use the diagram below for questions 13 - 18. Assume ∠C forms a right angle and AB∥JK.
13. Which angles represent a linear pair?
A. ∠AEF & ∠BEG
B. ∠F EH & ∠HEB
C. ∠ICD & ∠HEB
D. ∠JCH ∠DCE
E. none of the above
14. ∠AEF and ∠BEG are _________.
A. Supplementary
B. complementary
C. equivalent
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D. Vertical angles
E. both C and D
15. ∠JCD has measure of _____.
A. 90
B. 85
C. 180
D. 100
E. cannot be determined
16. Which postulate allows for the statement ∠F EH + ∠HEB = ∠F EB?
A. Segment Addition Property
B. Angle Addition Property
C. Midpoint Postulate
D. Vertical Angles Postulate
17. Suppose m∠BEG = 42. What is the measure of ∠AEG?
A. 138
B. 90
C. 42
D. 48
E. cannot be determined
18. E is the midpoint of AC. What does HI represent?
A. Bisector
B. midpoint
C. perpendicular
D. both A and C
E. none of the above
Answers: Standardized Test Prep Basics of Geometry
1. D
2. E
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3. B
4. A
5. D
6. B
7. A
8. B
9. A
10. B
11. E
12. D
13. D
14. E
15. C
16. B
17. A
18. D
1.2
Chapter 2
Quiz: Inductive Reasoning and Conditionals
Name:________________________ Hour:______ Date:______________
1. Find the next two terms in the following pattern: 2.4, 2.45, 2.456, 2.4567, . . .
2. Show by providing a counterexample that the following statement is false. “If you are
over 16, then you can drive legally.”
3. Consider the following conditional. Underline the hypothesis with one line and the
conclusion with two lines. “If a quadrilateral is a square, then it has four right angles and
four congruent sides.”
4. Consider the following conditional. “If an animal is a zebra, then it is a mammal.”
a. Write its inverse.
b. Write its converse.
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c. Write its contrapositive.
d. Assuming the original statement is true, which of the above (a − c) is also true?
5. Consider the following biconditional. “A triangle is scalene if and only if it has no sides
of equal length.”
a. Separate this into its two respective statements.
b. Do you think this is a true statement? Explain your reasoning.
Answers: Inductive Reasoning and Conditionals
1. 2.45678, 2.456789
2. Sample: You could be 80 and have your license revoked due to vision problems
3. “If a quadrilateral is a square, then it has four right angles and four congruent sides.”
4. a. If an animal is not a zebra, then it is not a mammal.
b. If an animal is a mammal, then it is a zebra.
c. If an animal is not a mammal, then it is not a zebra.
d. The contrapositive is true
5. a. If a triangle is scalene, then it has no sides of equal length. If a triangle has no sides
of equal length, then it is scalene.
b. Yes, this fits the definition of a scalene triangle.
Quiz: Deductive Reasoning and Algebraic Properties
Name:________________________ Hour:______ Date:______________
1. The following series of statements were created by the author Lewis Carroll, most noted
for Alice in Wonderland. What can you conclude using these clues?
a. Colored flowers are always scented.
b. I dislike flowers that are not grown in the open air.
c. No flowers grown in the open air are colorless.
2. Complete the following truth table:
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Table 1.1: (continued)
P
:p
p∧ : p
Table 1.1:
P
:p
p∧ : p
3. The following is the work shown to solve for variable x. Complete each step with its
appropriate property.
a. 6x + 32 = 16x + 98 Given
b. 6x + 32 + −6x = 16x + 98 + −6x ____________________
c. 32 = 10x + 98 ____________________
d. 32 + −98 = 10x + −98 ____________________
e. 56 = 10x
f.
56
10
=
10x
10
____________________
g. 5.6 = x ____________________
4. Match the correct statement to its property
a. If 3x = 16, then 16 = 3x
b. 16 = 16
c. If a = 3 and 3 = b, then a = b.
d. ∠C ∼
= ∠C
e. If 14a = 12, then 14a + 6 = 18.
Reflexive Property of Equality
Reflexive Property of Congruence
Symmetric Property of Equality
Symmetric Property of Congruence
Transitive Property of Equality
Transitive Property of Congruence
Addition Property of Equality
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Answers: Deductive Reasoning and Algebraic Properties
1. I like scented flowers.
2.
P
T
F
:p
F
T
p∧ : p
F
F
3. b. Addition Property of Equality
c. Simplification (Adding Like Terms)
d. Addition Property of Equality
f. Multiplication Property of Equality (Division Prop of Equality)
g. Simplification (Reflexive Property of Equality)
4. a. Symmetric Property of Equality
b. Reflexive Property of Equality
c. Transitive Property of Equality
d. Reflexive Property of Congruence
e. Addition Property of Equality
Quiz: Diagrams, Two-Column Proofs, and Congruence Theorems
Name:________________________ Hour:______ Date:______________
1. Write the following paragraph proof of the Congruent Supplements Theorem into twocolumn form. Include a sketch of the theorem. Congruent Supplements Theorem: If
two angles are supplements of the same angle (or of congruent angles), then the
two angles are congruent.
Using the definition of supplementary angles, m∠1 + m∠2 = 180 and m∠2 + m∠3 = 180.
Applying the substitution property of equality, m∠1 + m∠2 = m∠2 + m∠3. The Addition
Property of Equality allows us to subtract m∠2 from each side of the equation. Therefore,
m∠1 = m∠3 and ∠1 ∼
= ∠3.
2. What are some things you cannot assume when looking at a diagram?
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3. Prove the following in two-column format, “If two angles are congruent and supplementary,
then each is a right angle.
4. What are the categories of reasons (justifications) used in proofs?
5. Complete the proof of the following conditional All right angles are congruent.
Table 1.2:
Statement
Reason
∠M and ∠N are right angles
m∠M = 90 and m∠N = 90
c.
a.
b.
Substitution Property
Answers: Diagrams, Two-Column Proofs, and Congruence
Theorems
1.
Table 1.3:
Statement
Reason
∠1 and ∠2 are supplementary angles
m∠1 + m∠2 = 180
m∠1 + m∠2 = m∠2 + m∠3
m∠1 + m∠2 − m∠2 = m∠2 − m∠2 + m∠3
m∠1 = m∠3
Given
Definition of Supplementary Angles
Substitution Property
Addition Property of Equality
2. Right angles, parallel lines, perpendicular lines, and distance cannot be assumed from a
diagram - you must be told by notation.
3.
Table 1.4:
Statement
Reason
∠A and ∠B are congruent and supplementary
∠A ∼
= ∠B
m∠A + m∠B = 180
m∠A + m∠A = 180
2 m∠A = 180; m∠A = 90
m∠A and m∠B are right angles
Given
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Congruent Angles Theorem
Definition of supplementary angles
Substitution Property
Multiplication Property of Equality
Definition of a right angle
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Table 1.4: (continued)
Statement
Reason
4. Definition, postulates, previously proved theorems
5. a. Given
b. definition of a right angle
c. ∠M ∼
= ∠N
Chapter Test Reasoning and Proof
Name:________________________ Hour:______ Date:______________
1. Write the following statement as a conditional: All puppies are cute.
2. If is it snowing in Ohio, then it is not summer.
a. Write the converse of this statement.
b. Is the converse true? Explain your reasoning.
3. The measure of an angle is 3m. What is the measure of its complement?
4. Given the following clues, what can you conclude (if anything)? What reasoning allows
you to make this assumption?
a. Sonja is a freshmen.
b. Freshmen floss their teeth regularly.
5. Use a truth table to determine whether the following argument is valid:
a. If you invest in Ericon, then you will get rich.
b. You did not invest in Ericon.
c. Therefore, you did not become rich.
6. Solve the equation. Write the appropriate reasoning for each step.
11x + 43 = −23
7. What property justifies this statement: If AB = DM and AB = JK, then DM = JK.
8. Amphibians have skin that is scale-less and permeable to water.
a. Write the inverse of this statement.
b. Write the contrapositive.
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c. Write its converse.
d. Assuming the original statement is true, which of the above statements (a − c) is also
true?
9. Sketch a diagram that represents the given information. ∠SAP , an obtuse angle, is
←−→
bisected by DM .
10. Describe the following pattern and write the next four numbers in the sequence:
14, 5, 13, 4, 12, 3, . . .
11. Find a counterexample to disprove the hypothesis The sum of two irrational numbers is
irrational.
12. Find x, y, and the measures of each angle in the diagram below.
13. Prove the following: If AE ∼
= BE and DE ∼
= BE, then AC ∼
= BD
Answers: Chapter Test Reasoning and Proof
1. If an animal is a puppy, then it is cute.
2. a. If it is not summer in Ohio, then it is snowing.
b. No, it could be fall in Ohio and not snowing.
3. 90 − 3m
4. Sonja flosses regularly (Law of Detachment)
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5. The argument is invalid
6. 11x+43 = −23 Using the Addition Property of Equality, the equation becomes 11x = −66.
Using the Division Property of Equality, the equation becomes x = −6.
7. Transitive Property of Equality
8. a. If it is not an amphibian, then its skin is not scale-less nor permeable to water.
b. If something’s skin is not scale-less nor permeable to water, then it is not an amphibian.
c. If its skin is scale-less and permeable to water, then it is an amphibian.
d. Contrapositive
9.
10. 11, 2, 10, 1
11. π + −π = 0
12. X = 35, y = 18, 126◦ , 84◦
13. Using the transitive property of equality, AE ∼
= DE. This means all four segments are
congruent. Using the segment addition property, AC ∼
= BD
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Standardized Test Reasoning and Proof
Name:________________________ Hour:______ Date:______________
1. What is the next number in the sequence? 2, 4, 3, 6, 5, 10, . . .
a. 15
b. 20
c. 9
d. 19
2. What is the hypothesis of the statement I’ll have pizza if it is Friday night.
a. I’ll have pizza
b. Friday
c. It is Friday night
d. None of the above
3. Which property is stated here? If x = 3 and x = 4y, then 3 = 4y.
a. Reflexive Property of Congruence
b. Symmetric Property of Equality
c. Addition Property of Equality
d. Transitive Property of Equality
4. Which term is not considered an undefined geometric term?
a. Coplanar
b. Point
c. Plane
d. None of the above
5. Which of the following can be used to prove a statement?
a. Definitions
b. Postulates
c. Other theorems
d. all of the above
6. What is the converse of All cats have whiskers?
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a. If it is a cat, then it has whiskers
b. If it has whiskers, then it is a cat.
c. All whiskers belong to cats.
d. If it is not a cat, then it does not have whiskers.
7. Which is an instance of, If you are a band member, then you are a musician?
a. You are the bass player
b. You are a roadie
c. You are the stage manager
d. You are the busdriver
8. The measure of an angle is 6c. What is the measure of its supplement?
a. 90 − 6c
b. 180 − 6c
c. 45
d. cannot be determined
Answers: Standardized Test Reasoning and Proof
1. C
2. C
3. D
4. A
5. D
6. B
7. A
8. B �
1.3
Chapter 3
Quiz: Parallel Lines, Angles, and Transversals
Name:________________________ Hour:______ Date:______________
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1. Using the diagram below, list a pair of
a. Consecutive interior angles
b. Alternate interior angles
c. Corresponding angles
2. Two lines and a transversal form how many corresponding angles? What is true of these
angles if the lines are parallel?
3. Find the value of x to make lines a and b parallel.
4. Prove the following: If two lines and a transversal form same-side exterior angles that
are supplementary, then the lines are parallel.
Answers: Parallel Lines, Angles, and Transversals
1. a. Sample: ∠3 & ∠6
b. Sample: ∠3 & ∠5
c. Sample: ∠2 & ∠6
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2. 4 pairs of corresponding angles; they are congruent
3. x = 17
4. Suppose 2 lines are cut by a transversal forming supplementary consecutive exterior angles (1 and 2). ∠1 + ∠2 = 180. Angles 2 and 3 form a linear pair thus are supplementary.
So ∠2 + ∠3 = 180. Using the substitution property, ∠1 = ∠2. Angles 1 and 2 are corresponding angles. Using the corresponding angles postulate, because corresponding angles
are congruent, the lines are parallel.
Quiz: Equations of Lines
Name:________________________ Hour:______ Date:______________
1. Find the equation of the line passing through (−4, 8) with a slope of 43 .
2. Line m has a slope 54 . Line n has a slope of 45 . Are these lines perpendicular, parallel, or
neither.
3. Determine the slope of the line through (3, 7) and (−1.5, 19)
4. Consider a ladder. What reasoning allows you to conclude that the rungs are each
perpendicular to one side?
5. Prove the following: If a line is perpendicular to one of two parallel lines, then it is also
perpendicular to the other.
5. Write the symmetric statement for “is parallel to.” Is the statement true or false? Explain
your position.
6. What can you conclude using the following information? a ⊥ b, c ∥ d, b ⊥ c?
Answers: Equations of Lines
1. y = 43 x + 11
2. neither
3.
−8
3
4. The parallel to perpendicular theorem; because every rung is perpendicular to a side, the
rungs are parallel to each other
5. Suppose a line is perpendicular to one of two parallel lines. Then the measure of all
four angles equal 90. Because the lines are parallel, corresponding angles are congruent.
Therefore, the perpendicular of one is also perpendicular to the other.
6. If l is parallel to m, then m is parallel is l. True
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7. a is parallel to d.�
Quiz: Perpendicular Transversals and Non-Euclidean Geometry
Name:________________________ Hour:______ Date:______________
1. Explain why the following statement about the drawing is incorrect. a is perpendicular
to c.
−−→ −−→
2. In the diagram below CD ⊥ BD. Find y and the measure of each angle.
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3. State the Perpendicular Transversal Theorem.
4. Prove the following statement: If ∠5 and ∠6 form a linear pair and are congruent, then
the lines forming the angles are perpendicular.
5. Draw the taxicab circle with center (1, −3) and radius 3.
Answers: Perpendicular Transversals and Non-Euclidean Geometry
1. Simply because a is perpendicular to b, you cannot assume it is also perpendicular to c.
The diagram does not show this relationship.
2. y = 33, 39◦ , 51◦
3. If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to
the other.
4. According to the Linear Pair Theorem, linear pair angles are supplementary. Therefore,
angles 5 and 6 have a sum of 180. If they are congruent, using the substitution property,
2(m∠5) = 180 and using the division property of equality, ∠5 = 90. Thus, angle 5 is a right
angle and the lines are perpendicular.
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5.
Chapter Test Parallel and Perpendicular Lines
Name:________________________ Hour:______ Date:______________
1. Tell whether these two lines are skew, parallel, or perpendicular: y = 4x + 7 and y =
−1
x − 3.
4
2. Write an equation of the line with slope
2
3
containing point (1, −8).
3. Lines n and d are parallel. Could 40 and 40 be measurements of a pair of consecutive
interior angles? Explain your answer.
4. Suppose lines x and y are parallel. List:
a. A pair of vertical angles
b. A pair of corresponding angles
c. A linear pair
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5. Using the diagram above and x ∥ y. Suppose m∠8 = 98◦ . Find the measures of the
remaining seven angles.
6. Find the value of p that makes a ∥ b.
7. A passenger in a taxi wants to see how many distinct locations he can visit if the cab
travels two blocks without turning around. Plot the possible locations on the grid below.
8. Fill in the blank: The measure of a(n) _____ angle of a triangle is equal to the sum of
the measures of its opposite interior angles.
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9. Find the equation of the line parallel to y = 13 x − 2 passing through the point (−5, 5).
10. Elm Street is to be built parallel to Main Street. What is the value of x?
Answers: Chapter Test Parallel and Perpendicular Lines
1. Perpendicular
2. y = 23 x −
26
3
3. No, these angles must be supplementary
4. a. ∠5 & ∠7
b. ∠4 & ∠8
c. ∠5 & ∠6
5. Angles 2, 4, 6 all equal 98 and angles 1, 3, 5, 7 all equal 82
6. p = 70
7. Points should be located at the following coordinates: (4, 0), (5, 1), (3, 1), (2, 2), (6, 2), (3, 3), (5, 3), (4, 4)
8. Exterior
9. y = 31 x +
20
3
10. 110
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Standardized Test Parallel and Perpendicular Lines
Name:________________________ Hour:______ Date:______________
1. If two lines are cut by a transversal and corresponding angles are congruent, then the
lines are _______
a. Parallel
b. Intersecting
c. Congruent
d. None of the above
2. Two lines in which the product of slopes is −1 are _____.
a. Skew
b. Parallel
c. Perpendicular
d. None of these
2. The slope of the line formed by (−4, 5) and (8, −5) is:
a. 6/5
b. −5/6
c. 5/6
d. −6/5
A line with a zero slope is _____
a. Vertical
b. Horizontal
c. Skew
d. Oblique
5. Angles that lie on the same side of a transversal between two parallel lines are:
a. Congruent alternate interior angles
b. Supplementary consecutive interior angles
c. Congruent vertical angles
d. Congruent corresponding angles
6. Which of the following angles are alternate exterior angles?
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a. ∠1 and ∠8
b. ∠1 and ∠7
c. ∠2 and ∠5
d. ∠4 and ∠6
7. The sum of a pair of vertical angles is 170◦ . What is the measure of one angle?
a. 70
b. 100
c. 135
d. 85
8. Which is an equation for the line passing through (2, −3) with a slope of −2?
a. y = −2x + 1
b. y = −2x − 3
c. y = 2x − 3
d. x = 2
Answers: Standardized Test Parallel and Perpendicular Lines
1. A
2. C
3. B
4. B
5. B
6. B
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7. D
8. A
1.4
Chapter 4
Quiz: Triangle Sums and Congruent Figures
Name:________________________ Hour:______ Date:______________
1. MAPS ∼
= TOYS.
a. Draw a sketch of this situation.
b. ∠P ∼
= _____.
c. SY = _____.
d. If m∠A = 64◦ , then ______ = 64◦ .
2. Use the given coordinates to determine if △N OP ∼
= △CAT .
N = (2, −2), O = (5, 1), P = (4, 8)
C = (7, 5), A = (10, 8), T = (9, 13)
3. m∠1 = 32◦ and m∠3 = 44◦ of the triangle at the right. Find m∠2.
4. The sum of two exterior triangle angles is 277. What is the measurement of the remaining
exterior angle?
5. Define congruent figures.
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Answers: Triangle Sums and Congruent Figures
1. a. Answers may vary; point S is shared between both figures
b. SP
c. m∠O
2. No, OP ̸= AT
3. 104
4. 83
5. Geometric figures that have the same size and shape
Quiz: Triangle Congruence Using SSS, SAS, AAS, ASA, and
HL
Name:________________________ Hour:______ Date:______________
1. Are these two triangles congruent? If so, list the reasoning that allows you to make this
conclusion and write a congruency statement. If not, list the reasoning that would provide
a counterexample.
2. Using the diagram below, prove m∠A = m∠D.
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3. Assume AB = DE and AC = DF . What other piece(s) of information would you need
to prove these triangles congruent
a. Using SSS
b. Using SAS
c. Using ASA
4. Given AC ∼
= DC and ∠A ∼
= ∠D = 90◦ , prove △ABC ∼
= △DBC.
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Answers: Triangle Congruence Using SSS, SAS, AAS, ASA,
and HL
1. Yes, the triangles are congruent by ASA. △DEF ∼
= △CAB
2. ∠ECD ∼
= ∠BCA by the Vertical Angle Theorem. △ABC ∼
= △DEC by the ASA
Congruence Theorem. m∠A = m∠D by the CP CT .
3. a. BC = EF
b. m∠A = m∠D
c. m∠A = m∠D and m∠C = m∠F
4. BC ∼
= BC by the Reflexive Property of Equality. △ABC ∼
= △DEC by the HypotenuseLeg Congruence Theorem.
Quiz: Constructions, Isosceles Triangles, and Congruent Transformations
Name:________________________ Hour:______ Date:______________
1. List two types of congruent transformations.
2. Consider isosceles △JM K with vertex angle 102◦ . Find the measurement of the base
angle.
3. Are isosceles triangles always obtuse? Explain your reasoning.
4. Construct an equilateral triangle with side AB.
5. The measure of an exterior angle of an isosceles triangle is 110◦ . What are the possible
angles measures of the triangle? Explain your thought process.
6. Rotate ABCD 270◦ counterclockwise.
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Answers: Constructions, Isosceles Triangles, and Congruent
Transformations
1. Reflections and rotations
2. 39
3. No, 50 + 50 + 80 is an example of an acute isosceles triangle
4.
5. 70, 70, 40 if the exterior angle is of the base angle, 55, 55, 70 if the exterior angle is of the
vertex.
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6. The figure should have the following vertices: A′ = (−4, −4), B ′ = (−2, −2), C ′ =
(−2, 1), D′ = (−4, 1)
Chapter Test Congruent Triangles
Name:________________________ Hour:______ Date:______________
1. Draw a picture to represent △DOG ∼
= △M OP . Name all of the pairs of corresponding
congruent parts.
2. Draw a counterexample to the following statement: If two triangles have an AAA relationship, then the triangles are congruent.
3. Construct the perpendicular bisector of JM .
4. Consider isosceles triangle △JM K. Vertex ∠K measures 112◦ . Find m∠J.
5. Suppose △BAD has the following angle measures: m∠B = 65, m∠A = 57.5, m∠D =
57.5. What can you conclude about △BAD? What reason allows you to make your conclusion?
6. Which of the following does not yield a congruence transformation?
a. Rotating a chair in your room
b. Pushing your bike up the driveway
c. Enlarging a photograph
d. Your reflection in a mirror
7. Translate ABCD using the following rule: (x − 2, y + 0.5)
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8. Reflect △M N O over the y−axis.
9. Rotate △M N O 90 clockwise.
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Answers: Chapter Test Congruent Triangles
1. ∠D ∼
= ∠M, ∠O ∼
= ∠O, ∠G ∼
= ∠P, DO ∼
= M O, OG ∼
= OP , DG ∼
= MP
2.
3.
4. 34
5. Triangle BAD is isosceles with base angles A and D by the Converse of the Isosceles
Triangle Base Angles Theorem.
6. C
7. The image should have the following coordinates: A’ = (−1, 3.5), B’ = (.5, 1), C’ =
(−3, −3.25), D’ = (−1, .5)
8. The image should have the following coordinates: M ’ = (5, 5), N ’ = (−3, 2), O’ = (1, 6)
9. The image should have the following coordinates: M ’ = (5, 5), N ’ = (6, 1), O’ = (2, −3)
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Standardized Test Congruent Triangles
Name:________________________ Hour:______ Date:______________
1. Which of the following is a congruent transformation?
a. Dilation
b. Translation
c. Reflection
d. B and C only
2. A geometric construction uses the following materials:
a. Ruler
b. Protractor
c. Pencil
d. Compass
3. Two sides and an included angle of one triangle are congruent to two sides and an included
angle of a second triangle. These triangles are congruent by:
a. ASA Congruence Theorem
c. SAS Congruence Theorem
b. SSA Congruent Theorem
d. There is no postulate
4. Which of the following is not a triangle congruence postulate?
a. SSA
B. AAA
c. ASA
d. AAS
5. The sum of two consecutive interior angles of a triangle is 104◦ . The measure of the
opposite exterior angle is
a. 104
b. 84
c. 65
d. cannot be determined
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6. What additional information would you need to prove these triangles congruent using
ASA Congruence Theorem?
a. ∠W ∼
= ∠Z
b. T W ∼
= XZ
c. ∠T ∼
= ∠X
d. V W ∼
= YZ
7. A triangle with all three segments of 4 inches is:
a. Equilateral
b. Equilangular
c. A and B
d. none of the above
8. Point K = (3, 4). What is K ′ following a translation of 3 units right and 7 units up?
a. (0, −3)
b. (0, 11)
c. (6, 11)
d. (6, −3)
9. Suppose ∠A ∼
= ∠J, ∠B ∼
= ∠K, ∠C ∼
= ∠L. Which of the following is true?
a. △ABC ∼
= ∠KLJ
b. △ABC ∼
= △JKL
c. △ABC ∼
= △LKJ
d. none of the above
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Answers: Standardized Test Congruent Triangles
1. D
2. D
3. C
4. B
5. A
6. C
7. C
8. C
9. B
1.5
Chapter 5
Quiz: Midsegments, Circumcenters, and Incenters
Name:________________________ Hour:______ Date:______________
1. DE is a midsegment of △ABC. Find j.
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2. Draw M E with length 1.5” and construct its perpendicular bisector.
3. Construct an equilateral triangle with base length 2 cm.
4. Locate the circumcenter of triangle ABC.
Answers: Midsegments, Circumcenters, and Incenters
1. 22.5
2.
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3.
4.
D is the circumcenter
Quiz: Centroids, Orthocenters, and Triangle Inequalities
Name:________________________ Hour:______ Date:______________
1. X is the centroid of triangle ABC. Use the given information to find the value of m.
BX = 4m + 5 and BR = 9m.
2. Of the four types of points we have introduced (circumcenter, incenter, centroid, and
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orthocenter), when is each type inside the triangle?
3. Sketch and label the triangle with the given information: side lengths 3 cm, 7 cm, and
9 cm. Angle measures: 19◦ , 43◦ , and 118◦ . The longest side is on the left and shortest side
is on the bottom.
4. How can you organize a triangle using the following theorem: If one side of a triangle
is longer than another side, then the angle opposite the longer side is larger than the angle
opposite the shorter side?
Answers: Centroids, Orthocenters, and Triangle Inequalities
1. m =
5
2
2. The incenter is always inside the triangle, the circumcenter is inside the triangle when
the triangle is acute, the centroid is always inside the triangle, the orthocenter is inside the
triangle when it is acute.
3.
4. If you organize the information, the shortest side is opposite the smallest angle and the
longest side is opposite the largest angle.
Quiz: Indirect Proof
Name:________________________ Hour:______ Date:______________
1. What is the purpose of an indirect proof?
√
2. Prove 48 ̸= 6.9
3. Identify the two contradictory statements:
a. b and m are parallel
b. b and m have the same slope
c. b and m intersect
Prove: A right triangle cannot contain an obtuse angle.
Answers: Indirect Proof
1. To prove a statement by using a contradiction. Typically you use indirect proof when
you cannot prove using a series of justifications.
√
2. Assume 48 = 6.9. Then by squaring both sides, 48 = 47.61. however, this statement is
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not true. Therefore,
√
48 ̸= 6.9
3. Either A and C or B and C
4. Assume a right triangle has an obtuse angle. Then by the Triangle Sum Theorem,
90 + > 90 + m = 180. Simplifying the left side, you get a sum of greater than 180 degrees,
which is impossible. Therefore, a right triangle cannot have an obtuse angle.
Chapter Test Relationships Within Triangles
Name:________________________ Hour:______ Date:______________
1. List the angles of △ABC from smallest to largest. AB = 10, BC = 12, AC = 8.
2. List the sides of △M AP in order from longest to shortest. m∠M AP = 40◦ , m∠P M A =
110◦ .
3. Choose the correct term to complete the sentence: A (centroid, median) is a segment
whose endpoints are a vertex and the midpoint of the side opposite the vertex.
4. Fill in the blank with the correct term to complete the sentence. The ________ of a
triangle is the point of concurrency of the angles bisectors of the triangle.
5. Graph △DOG with vertices D(1, 3), O(−4, −4), G(2, −4). Find the coordinates of:
a. Its circumcenter
b. Its centroid
c. Its orthocenter
6. Find the value of j. D is the midpoint of AB and E is the midpoint of BC.
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7. Locate the incenter of △ABC.
8. Explain the relationship between the circumcenter, incenter, centroid, and orthocenter in
an isosceles triangle.
9. Are the following statements contradictory? Explain your position.
a. △ABC is acute
b. △ABC is scalene
10. Use indirect reasoning to prove quadrilateral ABCD cannot have four obtuse angles.
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11. Write a convincing argument that uses indirect reasoning. An obtuse cannot contain a
right angle.
Answers: Chapter Test Relationships Within Triangles
1. ∠B, ∠C, ∠A
2. AP , M P , AM
3. Median
4. Incenter
5. a matches to D, b matches to H, and c matches to I
6. j = 22.5
7. See picture
8. All lie on the same line the median connecting the vertex and base
9. These are not contradictory. A triangle can have acute angles and different side lengths.
10. Assume ABCD has four obtuse angles. Then by the Polygonal Sum Theorem, >
180 + > 180 + > 180 + > 180 = 360. However, the left side has a sum greater than
360 degrees. Therefore, ABCD cannot have four obtuse angles.
11. Assume an obtuse triangle has a right angle. Then by the Triangle Sum Theorem,
90 + x + y = 180. Using the Addition Property of Equality, x + y = 90. However, the
Addition to Inequality Property states that x < 90 and y < 90. This means that no angles
are obtuse. Therefore, an obtuse triangle cannot have a right angle.
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Standardized Test Relationships Within Triangles
Name:________________________ Hour:______ Date:______________
1. The point of concurrency of the angle bisectors of a triangle is called:
a. Centroid
b. Median
c. Orthocenter
d. Incenter
2. The midsegment joining a pair of sides of a triangle is _______ in relation to the third
side.
a. Parallel to
b. Twice as long
c. Half as long
d. A and C only
3. The point of concurrency of the perpendicular bisectors of a triangle is:
a. Circumcenter
b. Median
c. Centroid
d. none of the above
4. An altitude of a triangle can be:
a. The side of a triangle
b. Outside the triangle
c. In the interior of the triangle
d. All of the above
5. What is the negation of m ≥ 2?
a. m > 2
b. m < −2
c. m ≤ −2
d. m < 2
6. The orthocenter is the point of concurrency of:
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a. Angle bisectors
b. Medians
c. Altitudes
d. Perpendicular bisectors
Answers: Standardized Test Relationships Within Triangles
1. D
2. D
3. A
4. D
5. D
6. C
1.6
Chapter 6
Quiz: Polygonal Angles and Classifying Quadrilaterals
Name:________________________ Hour:______ Date:______________
1. A polygon has an interior angle sum of 1080 degrees. How many sides does the polygon
have?
2. Determine the measurement of an exterior angle of a regular decagon.
3. Draw a Venn diagram relating the following: quadrilateral, trapezoid, square, kite, and
rectangle.
4. True or false. A square is always a kite.
5. Find the value of g.
Answers: Polygonal Angles and Classifying Quadrilaterals
1. 8
2. 36 degrees
3. See sample below
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4. True
5. g = 112 degrees
Quiz: Parallelograms
Name:________________________ Hour:______ Date:______________
1. Given: ABCD is a parallelogram. Prove: ∠AEB ∼
= ∠DEC.
2. How do you find angle measurements in a parallelogram?
3. The measure of one interior angle of a parallelogram is 0.75 times the measure of another
angle. Find the measure of each angle.
4. In M AP S, M = 70, A = 110, P = 110. Is M AP S a parallelogram? Explain your position.
5. List the ways to prove a quadrilateral is a parallelogram.
6. Describe how to prove ABCD is a parallelogram. Assume AD = BC.
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Answers: Parallelograms
1. Because ABCD is a parallelogram, its opposite sides are parallel and congruent. Therefore, AB ∼
= DC. Diagonals in a parallelogram bisect each other, so AE ∼
= CE and
∼
BE = DE. By The SSS Triangle Congruence Postulate, △AEB ∼
= △DEC. And by
the CP CT, ∠AEB ∼
= ∠DEC.
2. Opposite angles in a parallelogram are congruent and consecutive angles are supplementary.
3. 77.14◦ and 102.86◦
4. M AP S is not a parallelogram. The order of the variables means A and P are consecutive
vertices. These two angles are equal not supplementary.
5. By showing: the diagonals bisect each other, an opposite pair of segments are parallel
and congruent, both pairs of opposite segments are parallel, both pairs of opposite segments
are congruent, both pairs of opposite angles are congruent.
6. By the reflexive postulate, segment AC is congruent to segment AC. By the SAS
congruence theorem, △DAC ∼
= △BCA. Using the CP CT , AB ∼
= DC. Therefore, both
pairs of opposite sides are congruent and ABCD is a parallelogram.
Quiz: Quadrilaterals and Biconditionals
Name:________________________ Hour:______ Date:______________
1. Using the vertices, determine if ABCD is a trapezoid. A = (−2, 4), B = (3, 4), C =
(−1, −5), D = (2, −5).
2. ABCD is a kite. Find m∠D.
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3. Is there enough information to classify the following figure as a square? Explain your
reasoning. You may assume JE = EL and KE = EM .
4. Using the vertices, name the figure as precisely as possible. Draw a sketch as a visual for
your answer. T = (−7, 4), I = (0, 9), P = (7, 4), S = (0, 0).
5. Can the following conditional be written as a biconditional? If so, write is as such. If
not, explain your reasoning. If a quadrilateral is a parallelogram, then its opposite sides are
congruent.
Answers: Quadrilaterals and Biconditionals
1. The slope of AB = 0. The slope of CD = 0. Since AB ∥ CD, ABCD is a trapezoid.
2. 67.5 degrees
3. No, using the given information, we can assume JKLM is a parallelogram. Using the
properties of a parallelogram, we can deduce all four angles are right angles. We also know
JK = M L and JM = KL, however, we do not know JK = JM .
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4. TIPS is a kite
5. Converse: If a quadrilateral has opposite sides that are congruent, then it is a parallelogram.
It can be written as a biconditional. A figure is a parallelogram if and only if its opposite
sides are congruent.
Chapter Test Quadrilaterals
Name:________________________ Hour:______ Date:______________
1. Using the figure below:
a. Find the sum of interior angles
b. Find m∠BCD.
2. Determine the measurement of an exterior angle of a regular octagon.
3. Create a hierarchy or Venn diagram relating the following terms: Polygon, triangle,
quadrilateral, isosceles trapezoid, parallelogram, square.
4. Fill in the blank with always, sometimes, or never: A parallelogram is _____________________
a kite.
5. When can a trapezoid be considered a parallelogram?
6. What is the most specific name for the figure below?
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7. Classify the shape on the coordinate plane below.
8. Consider the following definition: An inscribed polygon is a polygon whose vertices all lie
on a circle.
a. Write the converse of this conditional.
b. Write the biconditional of this definition
9. ABCD is an isosceles trapezoid with bases AD and BC. m∠D = 105◦ . Find m∠B.
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10. Given ABCD is an isosceles trapezoid with bases AD and BC, and m∠D = m∠C.
Prove ABCD is a rectangle.
Answers: Chapter Test Quadrilaterals
1. a. 1800
b. 150
2. 45
3.
4. Sometimes
5. A trapezoid is a parallelogram when it has two pairs of parallel sides.
6. Isosceles trapezoid
7. Slopes: AB = 1, AD =
−1
, BC
2
=
−2
, CD
3
= 1. ABCD is a trapezoid
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8. a. If a polygon’s vertices all lie on a circle, then it is an inscribed polygon
b. A polygon is inscribed in a circle if and only if all of its vertices lie on the circle
9. 75
10. Because ABCD is an isosceles trapezoid, its base angles are congruent. So ∠B ∼
= ∠C.
Furthermore, ∠D and ∠C are supplementary so ∠B = ∠C = ∠D = 90. ∠A and ∠B are
supplementary, therefore, ∠A = ∠B = ∠C = ∠D = 90. Therefore, ABCD is a rectangle
Standardized Test Quadrilaterals
Name:________________________ Hour:______ Date:______________
1. Which is not true of a parallelogram?
a. The diagonals bisect each other
b. It contains one pair of parallel sides
c. Opposite angles are congruent
d. Consecutive angles are supplementary
2. A quadrilateral with four congruent angles is called:
a. Rhombus
b. Square
c. Rectangle
d. Kite
3. Which statement about kites is true?
a. It has two sets of congruent angles
b. It has one pair of parallel sides
c. Its opposite sides are congruent
d. The diagonals are perpendicular
4. The diagonal of a parallelogram bisects its angle. Which of the following must it be?
a. Square
b. Rhombus
c. Rectangle
d. cannot be determined
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5. This type of figure has congruent base angles:
a. Trapezoid
b. Kite
c. Isosceles trapezoid
d. Rhombus
6. Which of the following does not always have perpendicular diagonals?
a. Trapezoid
b. Rectangle
c. Square
d. Rhombus
7. Three vertices of a kite are: A = (0, 5), B = (0, −1), C = (3, −2). Which could be the
fourth vertex?
a. (0, 2)
b. (3, 1)
c. (−2, 3)
d. (3, 2)
8. Which of the following conditionals is false?
a. A parallelogram is a square if it has four congruent sides
b. A quadrilateral is a rhombus if its diagonal bisects a pair of opposite angles
c. A figure is a kite if it has two pairs of consecutive congruent sides
d. A parallelogram is a rectangle if its diagonals are congruent
Answers: Standardized Test Quadrilaterals
1. B
2. C
3. D
4. B
5. C
6. A
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7. D
8. A
1.7
Chapter 7
Quiz: Ratios and Proportions
Name:________________________ Hour:______ Date:______________
1. The measures of DEF are in the extended ratio of 2 : 4 : 5. Find the measures of each
angle.
2. Solve for b :
8
b
= 57 .
3. The Department of Natural Resources is doing a study on the migration of deer. They
count 500 deer in 30 acres. Approximately how many deer are in the entire 600 acre forest?
4.
6
7
=
x
.
105
What are:
a. The means?
b. The extremes?
c. The value of x?
5. Two cities are 460 miles from each other. On a map, the distance between these cities is
20 inches. What is the scale of the map? �
Answers: Ratios and Proportions
1. 32.73, 81.82, 65.45
2. 11.2
3. 10, 000 deer
4. a. 6 and 105
b. 7 and x
c. 90
5. 23 miles per1 inch
Quiz: Similarity by AAA, SSS, SAS, and Similar Polygons
Name:________________________ Hour:______ Date:______________
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1. An original photograph 4” by 6” is to be enlarged by a scale factor of 75%. What are the
new dimensions?
2. KIDS ∼ JOHN . List all pairs of congruent angles and a ratio of similitude.
3. T U B ∼ SAY . T U = 21 cm and AS = 4.5 cm. What is the scale factor?
4. Are these two triangles similar? Explain your reasoning.
http://commons.wikimedia.org/wiki/File:Angle-Angle_Similarity_Postulate.png
5. Why is there an SSS Congruence Theorem and an SSS Similarity Theorem?
6. Karen’s mom is 64” tall and casts a 40” long shadow. Karen is 54” tall. How long will
her shadow be? �
Answers: Similarity by AAA, SSS, SAS, and Similar Polygons
1. 7” by 10.5”
2. ∠K ∼
= ∠J, ∠I ∼
= ∠O, ∠D ∼
= ∠H, ∠S ∼
= ∠N , ratio of similitude −KI/JO
3.
14
3
4. Yes, by the angle-angle similarity theorem, these are similar
5. Congruent figures are considered similar – they have congruent angle measures and the
ratio of similitude is 1.
6. 38.8” �
Quiz: Proportionality Relationships and Similarity Transformations
Name:________________________ Hour:______ Date:______________
1. Label the following scale factors as: enlargement, contraction, rotation.
a.
−1
2
b. 3
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c. 1.20%
d.
−9
2
2. Suppose JOKE has the following vertices. Find the image of each under
(0, 0), O = (−3, 6), K = (4, 12), E = (−1, 8)
(2
3
)
x, 23 y .J =
3. You are building a scale model of your high school’s baseball diamond. The length
1
between bases is 90 feet. If your scale is 30
, what will the distance be between bases?
4. You are making photo stickers for your friends’ yearbooks. Your photograph Is 3.3” by
3.3”. The printable area of the photo sticker is 1.1 inches by 1.1 inches. What is the scale
factor of the reduction?
Answers: Proportionality Relationships and Similarity Transformations
1. a. Rotation and contraction
b. Enlargement
c. Contraction
d. Enlargement and rotation
, 16
)
2. J’ = (0, 0), O’ = (−2, 4), K’ = ( 38 , 8), E’ = ( −2
3
3
3. 3 feet
4.
1
3
Chapter Test Similarity
Name:________________________ Hour:______ Date:______________
1. The following is a table relating the type of sandwich to the number sold.
a. What is the ratio of turkey sandwiches to pastrami on rye?
b. What is the ratio of peanut butter and jelly to all sandwiches sold?
Table 1.5:
Sandwich
Amount Sold
Ham and Cheddar
Turkey and Swiss
Pastrami on Rye
Roast Beef
15
12
8
7
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Table 1.5: (continued)
Sandwich
Amount Sold
Peanut Butter and Jelly
Veggie Wrap
Club Sandwich
2
9
19
2. Solve for c :
14
c
=
9
.
11
3. Is the following proportion true? Explain your answer.
4
5
=
64
90
4. Using the diagram below, find BC and list all the possible ratios of similitude.
5. True or false. All congruent figures are similar.
6. True or false. If a figure is similar, then it is also congruent.
7. Are the following triangles similar? Explain your rationale.
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8. Explain a fractal.
In questions 9 – 12, use kite ABCD below.
9. Draw the image of kite ABCD under a dilation of magnitude 1.5.
10. Prove that AB =
2
3
∗ (A’B’)
11. Prove CD∥C ′ D′ .
12. Which angle has the same measurement as ∠D?
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Answers: Chapter Test Similarity
3
2
1. a.
b.
1
36
2. c ≈ 17.11
3. The following proportion is not true because 4 ∗ 90 ̸= 5 ∗ 64
4. BC = 60 units; 30
, 40 , 45 . 60 , 2 , 3
45 60 30 40 3 2
5. True
6. False
7. Yes, by the AAA Similarity Theorem
8. A fractal is an object that is self-similar, meaning that if you enlarge one piece of the
object, it looks like the whole object
9. The image should have the following coordinates A’ = (−4.5, 3), B’ = (−6, 1.5), C’ =
(−3, −4.5), D’ = (−3, 1.5)
√
√
10. 12 + 12 = 32 ( 1.52 + 1.52 ). Both sides are approximately 1.414
11. Both lines have a slope of 3. Therefore, they are parallel
12. D’
Standardized Test Similarity
Name:________________________ Hour:______ Date:______________
1. Which of the following is not a triangle similarity theorem?
a. HL
b. AA
c. SSS
d. SAS
2. One serving of mashed potatoes calls for 13 cup milk. If one serving feeds two people, how
many cups of milk are needed for a party of 15?
a. 1 cup
b. 2 cups
c.
4
3
cups
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d. 2.5 cups
3. If, between two figures, sides lengths are proportional and angle measures are similar,
then the figures are:
a. Similar
b. Congruent
c. Neither
d. Cannot be determined
4. A rectangle is
5
16
as wide as it is long. How wide is the rectangle if it is 4 feet long?
a. 12.8’
b. 15”
c. 1.25”
d. 1’
5. Suppose under a dilation with scale factor 54 , AB = 23 cm. What is A’B’?
a. 23 cm
b. 21.75 cm
c. 18.4 cm
d. 28.75 cm
6. Which one is not a famous fractal?
a. Koch Snowflake
b. Pythagoras’ Curve
c. Sierpinski Triangle
d. Mandelbrot Set
7.
−3
4
does not represent which of the following:
a. Contraction
b. Rotation
c. Expansion
d. none of the above
8. If a line intersects two sides of a triangle and is parallel to the third line, then it divides
the two sides:
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a. Proportionally
b. Congruently
c. perpendicularly
d. none of the above
Answers: Standardized Test Similarity
1. A
2. D
3. C
4. B
5. D
6. B
7. C
8. A
1.8
Chapter 8
Quiz: Pythagorean Theorem and Geometric Mean
Name:________________________ Hour:______ Date:______________
√
1. Do the following lengths yield a right triangle? 4, 14, 2 53?
2. A rectangular parking lot measures
corner to its opposite?
1
4
mile by
5
8
mile. What is the distance from one
3. Find the geometric mean of 16 and 48.
4. Find the altitude of ABC.
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5. Do the following lengths yield an obtuse, acute, or right triangle? 12, 18, 20. Explain your
answer.
Answers: Pythagorean Theorem and Geometric Mean
√
1. Yes, 42 + 142 = (2 53)2
2. .673 mile
√
3. 16 3
4. 3.098
5. Acute angle 122 + 182 > 202
Quiz: Trigonometric Functions and Their Inverses
Name:________________________ Hour:______ Date:______________
1. Using ABC, find
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a. Sin(A)
b. Cos(A)
c. AC
2. A 16’ ladder is placed along a building 5’ from its base.
a. At what angle does the ladder make with the horizon?
b. How high along the building does the ladder reach?
3. An airplane is flying at an altitude of 35, 000’ and wants to make a smooth landing onto
the runway 150 miles away. At what angle should the plane descend?
4. What happens when you type sin−1 (1.1) into your calculator? Why do you think this
happens?
5. You are flying a kite with 50’ of string. The angle of elevation from the spool to the kite
is 65 degrees.
a. Draw and label a diagram to represent this situation.
b. What is the horizontal distance between you and your kite?
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Answers: Trigonometric Functions and Their Inverses
1. a.
b.
5
AC
4
AC
c. 6.42
2. a. 71.8
b. 15.2 feet
3. 2.5 degrees
4. Domain error; the sine values go between −1 and 1
5. 21.13’
Quiz: Acute/Obtuse Triangles
Name:________________________ Hour:______ Date:______________
1. Your triangle has a SAS situation. Which law, the Law of Cosines or the Law of Sines,
should you use? Explain your reasoning.
2. Samus wants to know how tall a tree is for his Biology project. He walks 100 feet away
from the base of a tree and uses an astrolabe to determine that the angle made from the
ground to the top of the tree is 33 degrees. The tree grows at an 85 degree angle from the
ground. Use this information to determine the height of the tree
3. Solve the following triangle. Draw a sketch to help you.
A =?
B = 65
C =?
a =?
b = 100
c = 89
Answers: Acute/Obtuse Triangles
1. The Law of Cosines because you have more sides than angles
2. 61.68’ tall
3. A = 62, C = 53, a = 97.42
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Chapter Test Right Triangle Trigonometry
Name:________________________ Hour:______ Date:______________
1. A rectangular pool has dimensions 15 feet by 25 feet. How long is its diagonal?
2. An isosceles triangle has a leg length 7 cm and base length 11 cm. What is its area?
3. What is the distance between (−6, −6) and (4, 13)?
4. Do these three segments represent an acute, right, or obtuse triangle: 5, 17, 22.89?
5. How can you identify a Pythagorean Triple?
6. Find the geometric mean of 9 and 81.
7. Consider a 30 − 60 − 90 degree triangle with shortest side 7 inches. Find the lengths of
the longer leg and hypotenuse.
√
8. An isosceles right triangle has hypotenuse 3 2 kilometers. What other information can
you conclude?
√
9. The triangle below has an altitude a = 4 5 cm. Find the measurements of legs m and n.
10. Using the triangle at the right:
a. Find tan(A)
b. Find measure of ∠A rounded to the nearest hundredth..
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11. The flagpole casts a 20 yard shadow. To the nearest whole yard, how tall is the flagpole?
12. Solve △ABC.
13. A 12−foot long wheelchair ramp must make a 5◦ angle with the horizon. How far away
from the base of the building will be the beginning of the ramp?
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Answers: Chapter Test Right Triangle Trigonometry
1. 29.15 feet
2. 23.82 cm2
3. 21.47 units
4. Acute, because 52 + 172 > 22.892
5. Sample: A Pythagorean Triple is a set of whole numbers that form a right triangle. For
example, 3, 4, and 5 represent a Pythagorean Triple because 32 + 42 = 52 .
6. 27
√
7. The longer leg is 7 2 inches and the hypotenuse is 14 inches.
8. The base angles are each 45 degrees and the each leg is 3 kilometers.
√
9. m = 12 cm; n = 6 5 cm
10. tan(A) = 7/12; b.∠A = 30.26 degrees
11. 35 yards
12. b = 46.88, c = 39.465
13. 11.95’
Standardized Test Right Triangle Trigonometry
Name:________________________ Hour:______ Date:______________
1. Consider a right triangle with legs j and k and hypotenuse l. Which of the following is
not true?
a. j 2 + k 2 = l2
b. j 2 − k 2 = l2
c. −j 2 − k 2 = −l2
d. k 2 = l2 − j 2
2. Which of the following is not enough information to solve a triangle?
a. Two angles
b. Two legs of a right triangle opposite angle
c. Two sides and an included angle
d. One side and the measure of its
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3. If the sum of the squares of two sides is less than the square of the longest third side,
then the triangle is
a. Right
b. Acute
c. Isosceles
d. Obtuse
4. Consider a 30 − 60 − 90 triangle. Suppose the shortest side is 3 cm. What is the measure
of the leg opposite the 60◦ angle?
a. 3 cm
b. 6 cm
√
c. 3 3 cm
√
d. 3 2 cm
5. Consider an isosceles right triangle. If the hypotenuse is
legs?
√
72”, what is the length of the
a. 6”
b. 72”
c. 8.5”
d. 36”
6. A 30−foot ladder leans against a wall, forming a 70◦ angle with the horizon. How far up
the wall does the ladder reach?
a. 8.24’
b. 28.19’
c. 10.26’
d. 15.67’
7. Which of the following is not used to solve for a non-right triangle?
a.
Sin(A)
a
=
Sin(C)
c
b. a2 = b2 + c2
c. a2 = b2 + c2 − 2(bc ∗ cos(A))
d. none of the above
8. Which of the following is the ratio of cosine(B)?
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82
a.
AB
BC
b.
AC
BC
c.
BC
AB
d.
AC
AB
Answers: Standardized Test Right Triangle Trigonometry
1. B
2. A
3. D
4. C
5. A
6. B
7. C
8. C
1.9
Chapter 9
Quiz: About Circles
Name:________________________ Hour:______ Date:______________
1. Identify the center and radius of the circle given the following equation (x+4)2 +(y)2 = 121.
2. Which of the following is a coordinate on the circle with equation (x − 2)2 + (y + 3)2 = 81
?
a. (0, 3)
b. (4, −1)
c. (2, 6)
d. None of the above
3. Write the equations for three concentric circles with center (1, 0).
4. Using the circle at the right, identify:
a. Diameter
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b. Radius
c. Secant
d. Tangent
e. Chord
5. Circle A has area 484π in2 . Circle B has diameter 44”. Are these circles congruent?
Explain your reasoning.
Answers: About Circles
1. Center (−4, 0) with radius 11
2. C
3. Sample: (x − 1)2 + y 2 = 2, (x − 1)2 + y 2 = 81, (x − 1)2 + y 2 = 9
4. a. CE
b.AE
←→
c. DC
←
→
d. D
←→
e. BC
5. Yes, by finding the area of circle B, you can show the circles are congruent
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Quiz: Tangents to Circles
Name:________________________ Hour:______ Date:______________
1. In the diagram, AC is a radius of circle A. Is DC tangent to the circle? Show your work
to illustrate your answer.
2. True or false. The distance between the centers of two circles is equal to the length of
the diameter of each circle.
3. DC passes through the center of circle A and F E is perpendicular to DC at point G.
The radius of the circle is 5 inches and AG = 4 inches. If GF = 3 inches, what is EF ?
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4. What is the perimeter of the pentagon pictured below?
Answers: Tangents to Circles
1. No, by the converse of Pythagorean’s Theorem, 1.52 + 2.672 ̸= 2.952 . Therefore, angle C
is not right and DC is not tangent to the circle
2. True
3. 6 inches
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4. 19.6 cm
Quiz: Arc Measures, Chords, and Inscribed Angles
Name:________________________ Hour:______ Date:______________
1. Minor arc XZ has measurement 99 degrees. What is the measurement of major arc
XY Z?
⊙
⊙
2. Suppose
R is congruent to
M . Given that arc AB is congruent to arc CD, what
can you conclude? (Draw a picture of the situation to help illustrate your solution)
3. Two concentric circles have radii 6 m and 12 m. A segment tangent to the smaller circle
is a chord of the larger circle. What is the length of the segment?
4. A 22” chord is 10” from the center of a circle. What is the radius of the circle?
5. The measure of arc BC is 125 degrees, which is intercepted by the inscribed angle ∠BDC.
What is the measure of the angle?
Answers: Arc Measures, Chords, and Inscribed Angles
1. 261
2. You can conclude that central angles ∠ARB and ∠CM D are congruent
3. 10.39”
4. 14.866”
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5. 62.5
Quiz: Angles and Segments of Chords, Secants, and Tangents
Name:________________________ Hour:______ Date:______________
Find the measure of each unknown.
1. Given: Diagram at the right, EC = 12, and BC = 26; determine: m∠CED and ED.
2. Given: Diagram at the right, DF = 6, F E = 12, F B = 10. Find m∠1 and x.
3. Given: Diagram at the right. Find m and m∠BF D.
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4. Explain the difference between a tangent segment and a secant segment.
Answers: Angles and Segments of Chords, Secants, and Tangents
1. ∠BED = 39, ED = 21.35
2. m∠1 = 51, x = 7.2
3. m∠BF D = 42, m = 4.5
4. A tangent segment touches a circle at one point while a secant segment intersects a circle
in two points.
Chapter Test Circles
Name:________________________ Hour:______ Date:______________
1. Using the circle below, label an example of the following:
a. Chord:
b. Secant:
c. Tangent:
d. Diameter:
e. Radius:
f. Central Angle:
g. Major arc:
h. Minor arc:
i. Semicircle:
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2. What is the equation for the circle at the right?
3. What is the diameter and center of the circle with equation (x + 4)2 + (y − 7)2 = 10 ?
4. Draw a vertical tangent to the circle below.
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90
5. Find the missing segments using the diagram below.
6. Find the distance between the centers of circles A and B.
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7. Determine the measures of as many arcs as you can. You can assume P L and QM are
diameters of circle O.
Answers: Chapter Test Circles
a. Chord: Sample: HG
←→
b. Secant: Sample: HG - the extension of chord HG
←→
c. Tangent: DE
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92
d. Diameter: AC
e. Radius: Sample - BC
f. Central Angle: Sample - ∠GBC
g. Major arc: Sample - arc AGH
h. Minor arc: Sample – arc ADH
i. Semicircle: arc ADC
2. (x + 4)2 + (y − 1)2 = 4
√
3. Center = (−4, 7) with diameter 2 10 units
4. Sample below
5. EB = 3 units, AD = 12 units, BF = 9 units, EF = 12 units, ED = 15 units, DF =
21 units
6. 5.60 units
7. Arcs: P O = 60, OL = 120, P M = 120, P M L = 180, P OL = 180
Standardized Test Circles
Name:________________________ Hour:______ Date:______________
1. A segment whose endpoints lie on the boundary of a circle is called a(n)
a. Tangent
b. Secant
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c. Chord
d. Diameter
2. In congruent circles, chords are congruent if
a. They have congruent arcs
b. They are parallel
c. They are proportional
d. none of the above
3. A circle has equation (x − 2)2 + (y + 7)2 = 36. What is its center?
a. (2, 7)
b. (0, 0)
c, (6, 6)
d. (2, −7)
4. A 14 cm chord is 6 cm from the center. What is the radius of the circle?
a. 9.21 cm
b. 3.61 cm
c. 6 cm
d. 85 cm
5. The diameter of a circle is 20 inches and a chord of the circle is 12 inches. How far is the
chord from the center of the circle?
a. 64 inches
b. 8 inches
c. 11.66 inches
d. 136 inches
6. A circle has center (11, 1) and radius
√
10. What is its equation?
a. (x + 11)2 + (y + 1)2 = 10
b. (x + 11)2 + (y − 1)2 = 10
c. (x − 11)2 + (y + 1)2 = 10
d. (x − 11)2 + (y − 1)2 = 10
7. In which case is the measure of an angle half of the sum of the intercepted arcs?
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a. Angles outside the circle
b. Angles on the circle
c. Angles inside the circle
d. Cannot be determined
8. The measure of arc AB = 167◦ . What is the measure of its inscribed angle?
a. 83.5
b. 167
c. Cannot be determined
d. None of the above
9. How many tangents do these circle share?
a. 0
b. 1
c. 3
d. 2
Answers: Standardized Test Circles
1. C
2. A
3. D
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4. A
5. B
6. D
7. C
8. A
9. D
1.10
Chapter 10
Quiz: Areas of Triangles, Quadrilaterals, and Similar Polygons
Name:________________________ Hour:______ Date:______________
1. The area of a triangle is 75 un2 and base length 15 units. What is the length of its
altitude?
2. Find the area of the trapezoid below.
3. Two pentagons have a ratio of similarity of 56 . The area of the smaller pentagon is 212 ft2 .
What is the area of the larger figure?
4. Compare and contrast the area formulas for parallelogram, kite, trapezoid, and rhombus.
How do each of these relate to the area of a triangle?
5. Why does it make sense that the units of area are expressed using the squared exponent?
6. Sketch the figure and determine its perimeter and area. The figure is a rhombus. Its side
length is 7 units and the length of one of its diagonals is 15 units.
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Answers: Areas of Triangles, Quadrilaterals, and Similar
Polygons
1. 10 units
2. 9.75 un2
3. 305.28 ft2
4. With the exception of the rhombus, all formulas involve the altitude and a base. Parallelograms are two triangles sharing their longest side.
5. Area involves the multiplication of two values. Each value has a unit. So unit∗unit = unit2
6. Perimeter = 28 units, area = 43.27 units
Quiz: Circumference, Arc Length, Area of Circles, and Area
of Sectors
Name:________________________ Hour:______ Date:______________
1. A pepperoni pizza has 14” diameter.
a. What is its circumference?
b. What is its area?
c. Suppose the pizza is cut into 8 slices.
i. What is the arc length of one slice?
ii. What is the area of 3 slices?
2. A monster truck tire is 42 inches in diameter. How many revolutions does the tire make
while traveling one mile?
3. The arc length of 120 degrees of a cinnamon roll is 6.28 inches. What is the radius of the
roll?
4. A circle is inscribed in a square. The square has a side length of 10 mm. What is the
area of the circle?
5. What is the area of a 57 degree sector of a circle with diameter 3 units? �
Answers: Circumference, Arc Length, Area of Circles, and
Area of Sectors
1. a. 14π ≈ 43.98 inches
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b. 49π ≈ 153.99 in2
c. i. 5.50 inches
ii. 51.33 in2
2. 480.18 revolutions
3. Approximately 3 inches
4. 25π ≈ 78.53 mm2
5. 1.12 un2
Quiz: Areas of Regular Polygons and Geometric Probability
Name:________________________ Hour:______ Date:______________
1. A regular heptagon has side length 42 cm and apothem 36 cm. Determine its area.
2. Find the area of a regular hexagon with radius 10 feet.
3. Define geometric probability.
4. Suppose school starts at 8:00 and ends at 4:00. You eat lunch at 12:00 p.m. If there is a
fire drill at a random time during the day, what is the probability it will happen after lunch?
5. A regular nonagon has side length 6 units and area of 788 un2 . What is the length of the
apothem?
6. Draw a sketch of the following situation and answer the question. Circles A and B are
concentric. Circle A has a radius 2.5 cm; circle B has diameter 30 cm. If you threw a dart,
what is the probability you will land in circle A?
Answers: Areas of Regular Polygons and Geometric Probability
1. 5, 292 cm2
2. 259.81 ft2
3. Probability that involves geometrical figures and/or quantities such as area or length
4.
1
2
or 50%
5. 29.19 un
6. 2.78%
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Chapter Test Perimeter and Area
Name:________________________ Hour:______ Date:______________
1. The length of a rectangle is 15 more than its width and encompasses an area of 356 in2 .
What is a possibility for the rectangle’s dimensions?
2. Find the area of the triangle below.
3. An isosceles trapezoidal table is sketched below. Find its area in square inches. (2.54 cm =
1 inch).
4. The area of a square is 196 mm2 . Determine its perimeter in centimeters.
5. The distance between Milwaukee, Minnesota and Phoenix, Arizona is 8.5” on a map. The
scale is 1 inch = 200 miles. What is the true distance between these two cities?
6. Two parallelograms have a ratio of perimeters of 65 . Suppose the perimeter of the smaller
parallelogram is 34 feet. What is the perimeter of the larger parallelogram?
7. A 16” diameter pizza has
a. _________________ circumference and ________________ area.
b. Suppose that pizza is cut into 10 slices. Determine:
i. The arc length of 3 slices of pizza.
ii. The area of 2 slices of pizza.
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8. The perimeter of a regular nonagon is 81y units. What is the length of a side?
9. An octagon with side length 5” has an apothem of 4.25”. Determine its area.
10. The distance between Acton and Barton is 30 miles, between Barton and Dayton is
95 miles, and between Canton and Dayton is 24 miles. What is the probability your car will
have a flat between Barton and Canton?
11. A rhombus has an area of 64 m2 . What is the product of its diagonals?
12. How many degrees does a minute hand of a clock travel from 12:00 to 4:00?
13. Suppose the length of the minute hand of a clock is 8 inches. How far would the minute
hand travel from 12:00 to 4:00?
Answers: Chapter Test Perimeter and Area
1. 12.804” wide and 27.804” long
2. 7.5 square units
3. 250.43 square inches
4. 56 mm
5. 1, 700 miles
6. 40.8 feet
7. a. 16π ≈ 50.265 inches, 64π ≈ 201.06 square inches
b. i. 15.0795 inches
ii. 40.21 square inches
8. 9y units
9. 85 square inches
10. 56.8%
11. 128 m
12. 1, 440 degrees
13. 201.06 inches
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Standardized Test Perimeter and Area
Name:________________________ Hour:______ Date:______________
1. What is the area of a regular pentagon whose apothem is 35.6 cm and side length is
47 cm?
a. 1673.20 cm2
b. 82.6 cm2
c. 8366 cm2
d. 4183 cm2
2. Which statement is false?
a. The height of the parallelogram is always inside the figure
b. Either pair of parallel sides can be used as the “base.”
c. If two parallelograms are congruent, then they have the same area.
d. If you translate a parallelogram in the coordinate plane, the area remains the same.
3. What is the length of the base of a triangle with altitude 9 cm and area 618 cm2 ?
a. 34.33 cm
b. 137.33 cm
c. 68.67 cm
d. Cannot be determined
4. What is the probability that you look at a clock and the second hand is between the 4
and 6?
a.
2
3
b.
1
3
c.
1
2
d.
1
6
5. Which is not true about probability?
a. It can be expressed as a fraction, percent, or decimal
b. It is a ratio of the wanted outcome to the entire possibilities]
c. It can be greater than 1
d. To express the probability of event B, you can write P (B)
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6. An orange has 3” diameter. What is its circumference?
a. 4.71 inches
b. 9.42 inches
c. 7.07 inches
d. 28.27 inches
7. An angle measures 100 degrees. What fraction is this of the circle?
a.
5
18
b.
1
3
c.
1
4
d.
5
8
8. What is the radius of a ball that rolls 210 feet in 50 revolutions?
a. 4.2 feet
b. 0.24 feet
c. 1.34 feet
d. 0.67 feet
1.11
Chapter 11
Quiz: Polyhedron and Representing Solids
Name:________________________ Hour:______ Date:______________
1. Define polyhedron.
2. Consider a pentagonal prism. Write the number of:
a. Faces:
b. Vertices:
c. Edges:
3. Use Euler’s Formula to determine the number of vertices regular dodecahedron.
4. Draw the orthographic view of the image below.
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commons.wikimedia.org/wiki/File:Birdhouse_030.jpg
Answers: Polyhedron and Representing Solids
1. A closed plane figure formed by three or more segments such that each intersects two
sides at their endpoints; a three dimensional figure made up of polygons.
2. a. 7
b. 10
c. 15
3. 30
4. Answers may vary
Quiz: Surface Area and Volume of 3-dimensional Figures
Name:________________________ Hour:______ Date:______________
1. A pencil holder is shaped like the figure below, with a 3” diameter and height of 4 inches.
How much metal would it take to make the sides and bottom, leaving the top open?
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2. How much cardboard does it take to make a cereal box 11.5” tall, 8.5” wide, and 3.5”
deep? How much cereal could it hold?
3. The Pyramid of Kafre is a right square pyramid is 144 meters tall with a base side length
of 215 meters. Determine its volume.
4. Sketch the solid and determine its surface area and volume. A right cone with diameter
of 18 cm and a slant height 25 cm.
5. What is the relationship between a prism and a pyramid with identical base and height?
Answers: Surface Area and Volume of 3-dimensional Figures
1. 44.77 in2
2. 335.5 square inches of cardboard to make; holds 342.13 in2
3. 2, 218, 800 m3
4. SA = 706.86 cm2 , V = 1978.92 cm3
5. The volume of a pyramid is one-third the volume of its corresponding prism
Quiz: Spheres and Similar Solids
Name:________________________ Hour:______ Date:______________
1. A yarn ball has diameter 4 inches. What is its volume?
2. How much material is needed to build a hollow globe with diameter 20 inches?
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3. The volume of a sphere is 288 cm3 . What is its diameter?
4. A hemispherical hanging planter has a radius of 7 inches. How much dirt can it hold?
5. Two similar prisms have heights 8 mm and 20 mm.
a. What is the ratio of similitude?
b. What is the ratio of their volumes?
c. How do their surface areas compare?
6. Rectangle prism A has dimensions l, w, and h. Rectangular prism B is twice as long and
twice as wide. How do their volumes compare?
Answers: Spheres and Similar Solids
1. 33.52 in3
2. 1256.64 in2
3. 8.19 cm
4. 718.38 in3
5. a. 2.5
b. 15.625
c. 6.25
6. The volume of prism B is 4 times that of prism A
Chapter Test Surface Area and Volume
Name:________________________ Hour:______ Date:______________
1. a. Name the figure below. Be as specific as possible!
b. List the values of the following:
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i. Edges:
ii. Faces:
iii. Lateral faces:
iv. Bases:
v. Lateral edges:
2. What is true about a polyhedron? How do polyhedra differ from prisms?
3. Draw the orthographic view of the doghouse illustrated below.
www.freeclipartnow.com/.../Doghouse.jpg.html
4. Draw the net of a truncated right pentagonal pyramid.
5. Find the surface area of the right hexagonal prism below.
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6. The barn below is to be painted with two coats of paint on all sides. If one gallon of paint
covers 150 ft2 , how many gallons of paint must you buy?
7. Draw a net of the below cylinder (does not have to be drawn to scale). What is its surface
area and volume of the cylinder?
8. How do the volumes of the two figures below compare?
9. According to Cavelieri’s Principle, two figures will have the same volume if ____________.
10. Find the surface area and volume of the pyramid below. Assume the length of the base
is 11 mm and its slant height is 18 mm.
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11. Label the following parts of the cone below.
a. Lateral edge
b. Radius
c. Altitude
d. Base
e. Determine its surface area and volume.
12. A basketball has a 9.4” diameter. How much leather is needed to make the basketball?
How much air can it hold?
13. A regular triangular pyramid has a volume of 114 ft3 and a surface area of 97 ft2 . How
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would the surface area and volume compare of a similar regular triangle pyramid with ratio
of similitude of 34 ?
Answers: Chapter Test Surface Area and Volume
1. a. Right pentagonal prism
b. i. Edges: 15
ii. Faces: 7
iii. Lateral faces: 5
iv. Bases: 2
v. Lateral edges: 5
2. Sample: A polyhedron is a three-dimensional figure in which all bases are equivalent. This
differs from a prism due to the fact that a prism has only two bases, or two equivalent bases.
3. See drawing for sample
4. See drawing for sample
5. 1, 716.66 un2 .
6. The answer is 21.208 gallons, but you must purchase 22 gallons.
7. See drawing for sample
8. See drawing for sample
9. The areas of their corresponding cross sections are equal in all cases.
10. Surface area = 517 mm2 ; volume = 691.21 mm3
11. a. Lateral edge letter I
b. Radius 3 units
c. Altitude 6 units
d. Base – circle
e. Determine its surface area and volume. SA = 63.22 un2 , V = 56.55 un3
12. SA = 277.59 in2 , V = 434.89 in3
( 3 )2
4
∗ 97 = 54.5625 ft2 and the volume of the
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13. The surface area of the smaller pyramid is
( )3
smaller pyramid is 43 ∗ 114 = 48.09375 ft3 .
Answers: Chapter Test Surface Area and Volume
1. Which equation does not represent Euler’s Formula?
a. F = E − V + 2
b. V = E − F + 2
c. E + V = F + 2
d. F + V = E + 2
2. The set of all points equidistant from a point in space is called
a. Circle
b. Plane
c. 3−dimensional geometry
d. Sphere
3. The intersection of two faces of a polyhedron is called
a. Edge
b. Base
c. Face
d. Vertex
4. Find the volume of a cylindrical drinking glass 11 inches tall with a 4 inch diameter.
a. 1520.53 in3
b. 138.23 in3
c. 553.92 in3
d. 150.8 in3
5. The surface area of a sphere is 916 m2 . What is its diameter?
a. 17.08 m
b. 8.53 m
c. 15.13 m
d. 30.27
6. A pyramid has 10 faces. What is the shape of its base?
a. Decagon
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110
b. Octagon
c. Nonagon
d. Cannot be determined
7. A polyhedron has 9 faces and 21 edges. How many vertices does it have?
a. 7
b. 9
c. 13
d. 14
8. A cylinder has a volume of 616 mm3 . What is the volume of its corresponding cone?
a. 616 mm3
b. 205.3 mm3
c. Neither A nor B
d. Cannot be determined
9.
10.
11.
12.
13.
1.12
Chapter 12
Quiz: Translations and Reflections
Name:________________________ Hour:______ Date:______________
1. Translate kite ABCD using the transformation vector (−2, 5).
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2. When can two matrices be multiplied together? What will be the product’s dimensions?
3. Organize the following into a matrix: 10 small t-shirts, 11 medium shorts, 0 large socks,
6 large shorts, 2 medium t-shirts, 12 medium socks, 4 small shorts, 8 large t-shirts, 9 small
socks.
4. Reflect △ABC over the x−axis and write the image points in a matrix.
5. True or false. The reflection of (a, b) over the line y = x, the image will have the
coordinate (−a, −b).
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Answers: Translations and Reflections
1. The image should have the following coordinates: A’ = (1, 6.5), B’ = (0, 5.5), C’ =
(2, 5.5), D’ = (1, 2.5)
2. When the “means” dimensions are the same value; the product’s dimensions will be the
“extremes”


10 4 9
3.  2 11 12
8 6 0
[
]
1
3 −2
4.
−4 −3 −3
5. False �
Quiz: Rotations and Compositions
Name:________________________ Hour:______ Date:______________
1. Write the matrix for R270 .
2.
] multiplication, determine the image of the following matrix under R180 .
[ Using matrix
0 4 −3
−6 11 2
3. Apply rx (ry (△ABC)). What single transformation does this equal?
4. A rotation of 810◦ is a composition of three rotations. Give one example to satisfy this
situation.
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5. Imagine riding a bicycle. What two transformations take place when the wheels move?
Is this an example of a composition? Explain your reasoning.
Answers: Rotations and Compositions
[
]
0 1
1.
−1 0
[
]
0 −4 3
2.
6 −11 −2
3. Rotation of 180 degrees
4. Sample: 360, 180, 270
5. Rotation and translation; the is not a composition because these two happen simultaneously; you do not apply one then the other
Quiz: Tessellations, Symmetry, and Dilations
Name:________________________ Hour:______ Date:______________
1. Explain why dilations are considered similarity transformations instead of congruent
transformations.
2. What is the rule for determining whether a regular polygon will tessellate the plane?
3. Using ABCD as a preimage, draw an example of tessellation.
4. How many symmetry planes does a pentagonal prism have?
[
]
−3 0 1
5. Consider YUM with coordinates
.
0 −5 4
a. Graph this polygon.
b. Apply S 4 .
5
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114
c. Is this an expansion, contraction, or rotation? Explain your position.
Answers: Tessellations, Symmetry, and Dilations
1. Dilations are enlargements and/or contractions, changing the segments lengths from the
preimage to the image. Congruent figures must have the same size and shape.
2. The sum of the interior angles must be a divisor of 360, meaning it must divide evenly
into 360
3. Answers may vary. Sample is shown below
4. 5
5. b.
[ −12
5
0
0
−4
4
5
16
5
]
c. Contraction, the image is smaller than the preimage
Chapter Test Transformations
Name:________________________ Hour:______ Date:______________
1. The transformation vector is (0, −3). Suppose M = (0, −9). Find M ’, the image of M .
2. Suppose the point (2, 3), lying in Quadrant I. Give a vector that will translate this point
into:
a. Quadrant III.
b. Quadrant IV.
3. Write the coordinates of the figure below in a point matrix.
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[
]
[
]
−1 4
7 0 −5
4. [A] =
. [B] =
. Does the sum [A] + [B] exist? Explain your answer.
5 7
2 10 8
5. Using the above matrices, which product exists, AB or BA? Explain your reasoning.
Find the product matrix.
6. Reflect the triangle over the y−axis and write the image points in a matrix.
7. Rotate the kite 90◦ counterclockwise and write the image points in a matrix.
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8. Match the following matrices to the appropriate rotation.
[
]
−1 0
a.
- i. R270
0 −1
]
[
0 −1
- ii. R180
b.
1 0
]
[
1 0
- iii. R90
c.
0 1
[
]
0 1
d.
- iv. R360
−1 0
9. Suppose rf (re (MATH)) using the figure below.
a. What type of isometry is formed?
b. Suppose the distance between lines e and f is 9 cm. What is the measurement of the
above isometry?
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10. Can a regular heptagon tessellate the plane? Explain your answer.
11. Outline and color the template of this tessellation.
integratedacademics.wikispaces.com
12. What type of symmetry does this butterfly possess? Draw in all symmetry lines.
http://www.clker.com/clipart-11571.html
13. How many planes of symmetry does a soup can possess?
14. True or false: A rectangle has both reflection symmetry and rotational symmetry.
15. True or false: Every figure that has reflection symmetry has rotational symmetry.
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16. The following represents a toy store inventory. Write the new matrix after an increase
of 150%.


4 1 7
2 9 3
0 8 6
17. Apply the following dilation to the triangle below: S 3 .
2
Answers: Chapter Test Transformations
1. M ’ = (0, −12)
2. a. Quadrant III. Sample: v = (x − 10, y − 4)
b. Quadrant IV. Sample: v = (x + 2, y − 8)
[
]
3 2 −3 4
3.
1.5 .5 −2.5 .5
4. Sample: The sum does not exist because the dimensions of matrix A do not equal the
dimensions of matrix B. Matrix addition adds cells together, thus the dimensions must be
equal.
5. Sample: The product
[ BA exists]because the “inside” or means dimensions are the same.
1 40 37
The product of BA =
49 70 31
[
]
−1 2 −3
6.
4 3 3
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[
]
−6 −5 −2 −5
7.
3
2
3
4
8. a-ii
b-iii
c-iv
d-i
9. a. Translation
b. 18 cm
10. Sample: A regular heptagon cannot tessellate the plane because any one interior angle
has a measure 128.57, which is not evenly divisible into 360.
11. The original region is a nonconvex irregular pentagonal region
12. Sample: This butterfly possesses reflection symmetry, vertically through the body.
13. Sample: A soup can possesses an infinite amount of vertical symmetry planes and one
horizontal symmetry plane.
14. True
15. False, see

6 1.5

16. 3 13.5
0 12
as acceptable.
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butterfly above for counterexample

10.5
4.5  Students may round up the fractional answers; this will be your choice
9
Due to the context of the question, whole number answers may be acceptable.
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[
]
3 7.5 9
17. The new triangle should have the following vertex locations:
3 4.5 1.5
Standardized Test Transformations
Name:________________________ Hour:______ Date:______________
1. Which of the following is the rotation of 90◦ matrix?
[
]
0 −1
a.
1 0
[
]
−1 0
b.
0 −1
[
]
0 −1
c.
−1 0
]
[
1 0
d.
0 1
2. What is the image of X = (4, −6) under (x − 9, y + 2)
a. (5, 8)
b. (−5, 8)
c. (−5, −4)
d. (5, −4)
3.
a.
b.
c.
[
]
]
3 2 [
1 2
Multiply
−1 .5
[
]
1 3
[ ]
1
3
[
]
4 4
0 2.5
d. Cannot be multiplied
4. Which statement is true?
a. The reflection over the line y = x is the same as a rotation of 90 degrees
b. A figure may have rotational symmetry but not reflection symmetry
c. A rotation of 180 is a reflection over the x axis following a rotation of 90 degrees
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d. If (x, y) is reflected over the y−axis, the image is (x, −y)
5. rx (ry (A)) implies
a. Reflecting A over the x axis then the y axis
b. Reflecting A over the y = x line
c. Reflecting A over the y axis then the x axis
d. Rotating A 90 degrees
6. A regular hexagon has _____-fold rotation symmetry
a. 6
b. 7
c. 2
d. none
7. Which of the following letters possess two lines of symmetry?
a. T
b. O
c. J
d. M
8. Which figure has rotational symmetry but not reflection symmetry?
a. Rhombus
b. Trapezoid
c. Kite
d. Parallelogram
Answers: Standardized Test Transformations
1. A
2. C
3. D
4. B
5. C
6. A
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7. B
8. D
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