Name class date Geometry Unit 1 Practice...H K L P e. T P S G 2.What is a correct name for this...

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1© 2015 College Board. All rights reserved. SpringBoard Geometry, Unit 1 Practice

LeSSon 1-1 1. Persevere in solving problems. Identify each

figure. Then give all possible names for the figure.

a.

Q

T

S

b.

AB

C

c. D

Fm

d. H K L P

e. T

P S

G

2. What is a correct name for this plane?

W

Z

R

X

A. plane XZR

B. plane WZ� ��

C. plane WXZ

D. plane X

3. How many different rays are in the figure? Name them.

Y

W

X

Z

Geometry Unit 1 Practice

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© 2015 College Board. All rights reserved. SpringBoard Geometry, Unit 1 Practice

4. How many different angles are in the figure? Name them.

T

W

Z

Y

X

5. Use appropriate tools strategically. Draw each figure.

a. RS

b. TV� ��

c. WX� ��

d. plane T containing points D and E

e. ∠CDE

LeSSon 1-2 6. Use this diagram.

B

C

PD

A

a. How many radii are shown? Name them.

b. How many diameters are shown? Name them.

c. How is a chord similar to a diameter? How is a chord different from a diameter?

d. Make use of structure. Suppose you have two different diameters of a circle. What must be true about the point where the diameters intersect?

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7. Two angles are complementary. They also have the same measure. Which statement is correct?

A. The two angles are obtuse.

B. The two angles are acute.

C. The two angles must be adjacent.

D. The two angles cannot be adjacent.

8. Reason quantitatively. ∠P and ∠Q are supplementary. The measure of ∠P is 30 degrees more than the measure of ∠Q. What is the measure of each angle?

9. This diagram shows lines PQ� ��

, RS� ��

, and TV� ��

. How many angles appear to be obtuse? Name them.

P

R

M

T

N

Q

V

S

10. Think about a chord of a circle and a radius of the same circle.

a. What two things do the chord and the radius have in common?

b. How are the chord and the radius different?

LeSSon 2-1 11. Use inductive reasoning to determine the next two

terms in each sequence.

a. 12, 17, 22, 27, . . .

b. 12, 17, 27, 42, . . .

c. 12, 17, 23, 30, . . .

d. 12, 60, 300, 1500, . . .

e. 1, 4, 9, 16, . . .

12. The fourth term of a sequence is 40. Write the first five terms of two different sequences that satisfy that condition.

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13. Persevere in solving problems. Use this picture pattern.

a. Draw the next two shapes in the pattern.

b. What numbers represent the next three figures in the pattern?

c. Verbally describe the pattern of the sequence.

d. How many dots are added from the first diagram to the second? From the second diagram to the third? From the third to the fourth? Explain how to find the nth term.

14. Which rule describes how to find the next term in the sequence?

0, 3, 9, 21, 45, 93, . . .

A. Multiply the previous term by 3.

B. Add 3 to the previous term, and then multiply the result by 3.

C. Multiply the previous term by 2, and then add 3.

D. Divide the previous term by 3, and then add 3.

15. Construct viable arguments. Explain how you knew that the rules you did not choose in Item 14 were incorrect.

LeSSon 2-2 16. Use expressions for even integers to show that the

product of two even integers is an even integer.

17. Consider these true statements.

• Allglassobjectsarebreakable.

• Allwindshieldsaremadeofglass.

• Tonette’scarhasawindshield.

Based on deductive reasoning, which of the following statements is not necessarily true?

A. Tonette has a breakable object.

B. Tonette has a glass object.

C. All windshields are breakable objects.

D. All breakable objects are glass.

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18. Make use of structure. Use deductive reasoning to prove that x 5 5 is not in the solution set of the inequality 2x 1 1 , 7. Be sure to justify each step in your proof.

19. During the first month of school, students recorded each day on which they had a quiz in math class. A student stated that there is a math quizeveryTuesdaymorning.Isthestudent’sstatement a conjecture or a theorem? Explain.

20. Reason abstractly. A student knows that (1) any two diameters in a circle bisect each other and (2) RS and TV are two different diameters in the same circle. The student concludes that RS and TV bisect each other.

a. Is this an example of inductive or deductive reasoning? Explain.

b. Is the conclusion correct? Support your answer.

LeSSon 3-1 21. Make use of structure. In each statement, tell

whether each bold term is undefined or defined.

a. An angle is formed by two rays that have a common endpoint.

b. A line segment consists of two points and all the points between them.

c. A triangle is the union of three segments that intersect at their endpoints.

d. If two lines intersect, then there is exactly one plane that contains the two lines.

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22. Model with mathematics. Complete this two-column proof by providing the reasons for each statement.

Given: x3 52

72

52

Prove: x 5 23

Statements Reasons

1. x3 52

72

52 1. a.

2. 3x 2 5 5 214 2. b.3. 3x 5 29 3. c.4. x 5 23 4. d.

23. Suppose you are given that p 5 2q 1 1 and that p 5 8. Which of the following statements can you prove?

A. p 1 1 5 2q

B. p 1 q 5 9

C. 2q 1 1 5 8

D. p 1 8 5 q

24. Identify the property that justifies the statement: If 3x 5 221, then x 5 27.

A. Addition Property of Equality

B. Distributive Property

C. Division Property of Equality

D. Transitive Property of Equality

25. a. Write a geometric statement that does not use any defined terms in geometry.

b. Write a geometric statement that does not use any undefined terms.

LeSSon 3-2 26. Reason abstractly. Write each statement in

if-then form.

a. The only time I wake up early is when I set my alarm clock.

b. I eat breakfast at a restaurant only if it is a weekend.

c. An obtuse angle has a measure between 90° and 180°.

27. State or describe a counterexample for each conditional statement.

a. If x2 5 25, then x 5 5.

b. If three points A, B, and C are collinear, then B is between A and C.

c. If a triangle is obtuse, then it cannot be isosceles.

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28. Suppose that this statement is true: If I wear boots or a raincoat, then I carry an umbrella. Also suppose that the hypothesis of that statement is true. Which statement must also be true?

A. I am wearing boots.

B. I am not wearing a raincoat.

C. I am not carrying an umbrella.

D. I am carrying an umbrella.

29. Write a true conditional statement that includes

this hypothesis: x3 18

21

5 .

30. Model with mathematics. Write a two-column proof to prove that your conditional statement in Item 29 is true.

Statements Reasons

1. x3 18

21

51. Given

2. a. 2. b.3. c. 3. d.4. e. 4. Division (or Multiplication)

Property of Equality

LeSSon 3-3

31. Write the inverse and the contrapositive of each statement.

a. If it is raining, then I stay indoors.

b. If I have a hammer, then I hammer in the morning.

32. Write the following biconditional statement as two conditional statements:

People have the same ZIP code if and only if they live in the same neighborhood.

33. Make use of structure. Use this statement: If 3x 5 0, then x fi 0.

a. Is the statement true? Explain.

b. Write the converse of the statement, and explain whether or not the converse is true.

c. Write the inverse of the statement, and explain whether or not the inverse is true.

d. Write the contrapositive of the statement, and explain whether or not the contrapositive is true.

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34. Which two forms of a conditional statement always have the same truth value?

A. statement and inverse

B. inverse and contrapositive

C. converse and contrapositive

D. converse and inverse

35. Reason abstractly. Use this statement: If two lines form equal adjacent angles, then the lines are perpendicular. Then tell whether each statement is the inverse, converse, or contrapositive of the original statement.

a. If two lines are not perpendicular, then they do not form equal adjacent angles.

b. If two lines do not form equal adjacent angles, then they are not perpendicular.

LeSSon 4-1 36. Suppose point T is between points R and V on a

line. If RT 5 6.3 units and RV 5 13.1 units, then what is TV?

A. 2.5 units

B. 6.8 units

C. 7.8 units

D. 19.4 units

37. Suppose P is between M and N.

a. If MN 5 10, MP 5 x 2 1, and PN 5 x 1 1, what is the value of x?

b. If PM 5 2x 2 5, PN 5 6x, and MN 5 5x 1 4, what is the value of x?

38. Attend to precision. Use the centimeter ruler shown.

14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

R W

a. What is the length of RW?

b. What number on the ruler represents the midpoint of RW?

c. Suppose Q is a point on RW� ��

. If QW 5 12, what are the possible coordinates of point Q?

d. Suppose point T is between points R and W and RTTW

12

5 . What is the length of RT?

39. Reason quantitatively. On a number line, the coordinate of point A is negative and the coordinate of point B is positive.

a. When will the midpoint of AB be positive?

b. When will the midpoint of AB be negative?

c. When will the midpoint of AB be zero?

d. When will the distance from A to B be negative?

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40. Points P, M, and T are on a line and PT 2 PM 5 MT. Which point is between the other two? Explain your answer.

LeSSon 4-2 41. Make sense of problems. Suppose that PQ

� �� bisects

∠MPN . What conclusion can you make?

42. Suppose AT� ��

bisects ∠CAR. If ∠m CAT 5 5x and ∠m CAR 5 9x 1 7, what is ∠m TAR ?

43. Suppose two angles are supplementary. Which of the following terms CANNOT describe both angles?

A. acute

B. adjacent

C. congruent

D. vertical

44. ∠D and ∠E are complementary. If ∠m D 5 5x 1 3 and ∠m E 5 3x 2 1, what is x?

45. a. Attend to precision. Draw a single diagram to represent the statements shown.

Obtuse angle PQR is bisected by QA� ��

.

∠PQA is bisected by QB� ��

.

∠BQA is bisected by QC� ��

.

∠AQR is bisected by QD� ��

.

b. Suppose ∠m PQR 5 128°. Find m AQR∠ .

LeSSon 5-1 46. Which expression represents the distance between

points (m, n) and (p, q)?

A. m n p q( ) ( )2 22 1 2

B. m p n q( ) ( )2 21 1 1

C. m p n q( ) ( )2 22 1 2

D. m p n q( ) ( )2 22 1 2

47. A segment has endpoints L(22, 7) and K(5, 23). What are the coordinates of the midpoint of LK ?

48. Attend to precision. The coordinates of the vertices of a triangle are A(24, 6), B(4, 22), and C(26, 24).

a. Find AB.

b. Find BC.

c. Find AC.

d. Based on the lengths of the sides, what kind of triangle is ABC� ?

49. The coordinates of the vertices of a triangle are D(5, 6), E(7, 5), and F(4, 3). Find the perimeter of the triangle.

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50. Model with mathematics. Every point on a circle is the same distance from the center of the circle. If (x, y) represents any point on a circle and (5, 2) is the center of the circle, use the Distance Formula to represent the length of the radius r of the circle.

LeSSon 5-2 51. Model with mathematics. BC has endpoints

B(23, 25) and C(12, 12). Find the coordinates of the midpoint of BC .

52. In the diagram shown, points S and T are the midpoints of PQ and PR, respectively.

x

y

2

4

6

8

24

22

222426 2 4 6 108

P(4, 7)

R(10, 3)

Q(22, 23)

S

T

10

26

a. Find the coordinates of points S and T.

b. Find the length of ST .

53. Construct viable arguments. Given: A(2, 5), B(0, 0), and C(4, 2).

a. Find the coordinate of M, the midpoint of BC.

b. Which of the points, B, M, or C, is closest to A?

54. Which expression represents the midpoint of the line segment with endpoints (x, y) and (p, q)?

A.

x y p q2

,2

1 1

B.

x p y q2

,2

2 2

C.

x p y q2

,2

1 1

D.

xp yq2,2

55. For the coordinates (5, 8) and (9, 14), one is an endpoint of a line segment and the other is the midpoint. How many possibilities are there for the other endpoint? Find each one. Explain your method.

LeSSon 6-1 56. Construct viable arguments. Use the diagram

shown.

TC

A

B

1

2

Write a statement that can be justified by each of the following:

a. definition of angle bisector

b. Angle Addition Postulate

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57. Use the diagram shown.

P Q R

a. What is the justification for the statement that PQ 1 QR 5 PR?

b. Suppose Q is the midpoint of PR. What is the justification for the statement that PQ 5 QR?


B C

AD

2x 1 5

Suppose m ABC∠ 5 90° and ∠m ABD 5 2x 1 5.

a. Write a statement that can be used as a justification that ∠m DBC 5 90 2 (2x 1 5).

b. Write a justification for the statement that ⊥AB CB.

59. What can you use to prove that B is the midpoint of AC?

A

B

y

x

C

A. ruler

B. protractor

C. definition of midpoint

D. folding AC on point B

60. Reason abstractly. Which statement CANNOT be justified by the use of the Distance Formula?

x

y

2

4

6

24

22

222426 2 4A(0, 0)

C(5, 5)D(0, 5)B(25, 5)

6

26

A. AB 5 AC

B. D is the midpoint of BC .

C. ⊥BC AD

D. AD 5 DC

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LeSSon 6-2 61. Consider the diagram and the given statements for

a proof.A

B C

ED

Statements Reasons

1. D is the midpoint of AB .

2. AD 5 DB

3. AD > DB

Which of the following could be the correct Given and Prove statements?

A. Given: AD > DB; Prove: AD 5 DB.

B. Given: AD > DB; Prove: D is the midpoint of AB .

C. Given: D is the midpoint of AB ; Prove: E is the midpoint of AC .

D. Given: D is the midpoint of AB ; Prove: AB > DB.

62. Make sense of problems. Complete the proof.

A B

E

3

21

C

D

Given: ∠1 is supplementary to ∠2; � ��BD bisects

∠CBE.

Prove: ∠1 is supplementary to ∠3.

Statements Reasons

1. � ��BD bisects ∠CBE . 1. Given

2. ∠2 >∠3 2. a.

3. m∠2 5 m∠3 3. Definition of congruent angles

4. ∠1 is supplementary to ∠2.

4. b.

5. m∠1 1 m∠2 5 180 5. Definition of supplementary angles

6. m∠1 1 m∠3 5 180 6. c.7. d. 7. e.

63. Complete the proof.

Given: m∠1 5 37; m∠PTR 5 53

Prove: m∠2 5 16

Statements Reasons

1. a. 1. Given2. m∠1 1 m∠2 5

m∠PTR2. b.

3. 37 1 m∠2 5 53 3. Substitution4. c. 4. d.


V T

S

R

12

Given: m∠RVT 5 2(m∠1)

Prove: � ��VS bisects ∠RVT.

T

R

Q

P

1

2

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Statements Reasons

1. m∠RVT 5 2(m∠1) 1. Given2. a. 2. Angle Addition

Postulate3. 2(m∠1) 5 m∠1 1

m∠23. b.

4. m∠1 5 m∠2 4. Subtraction Property of Equality

5. ∠1 > ∠2 5. c.

6. d. 6. e.

65. Critique the reasoning of others. A student says that the statement below can be justified by the definition of complementary angles.

If ∠A and ∠B are both complementary to ∠T, then ∠A > ∠B.

Isthestudent’sreasoningcorrect?Explain.

LeSSon 7-1 66. Use appropriate tools strategically. Use the

protractor shown.180

170160

150

140

130

120110

100908070

60

50

40

3020

100 18017

016

0150140

130 1

20110

100 90 80 70 6050

4030

2010

0

BA

CD

a. Find m∠BAC.

b. Find m∠BAD.

c. Find m∠CAD.

d. Suppose � ��AP bisects angle ∠DAC. At what

degree measure will � ��AP lie on the protractor?

67. Describe the relationship between each pair of angles.

213 4

5 6

7 8

a. ∠1 and ∠5

b. ∠4 and ∠5

c. ∠7 and ∠6

d. ∠4 and ∠6

68. In the diagram shown, lines , and m are parallel.

1 2

l m

p65

3 487

Which pair of angles does NOT represent corresponding angles?

A. ∠1 and ∠4

B. ∠5 and ∠7

C. ∠6 and ∠8

D. ∠2 and ∠4

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69. Attend to precision. Suppose that ∠1 and ∠2 are same-side interior angles formed by two parallel lines cut by a transversal, and that m∠1 5 7x 2 4 and m∠2 5 20x 2 5.

a. What is the value of x?

b. What is m∠1?

c. What is m∠2?

d. Explain how you found your answers.

70. Complete the proof that if parallel lines are cut by a transversal, then same-side exterior angles are supplementary.

1 2m

n

t

43

5 687

Given: m || n

Prove: m∠7 1 m∠1 5 180

Statements Reasons

1. m || n 1. a.

2. ∠3 > ∠7, ∠1 > ∠5 2. b.

3. m∠3 5 m∠7, m∠1 5 m∠5

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

4. m∠3 1 m∠5 5 180 4. c.5. d. 5. Substitution

Property of Equality

LeSSon 7-2 71. Use the diagram shown.

1 2 m

n

t

43

5 687

a. Suppose m∠5 5 130°. What is m∠3 so that m || n?

b. Suppose m∠8 5 141°. What is m∠4 so that m || n?

c. Suppose m∠3 5 42°. What is m∠6 so that m || n?

d. Suppose m∠7 5 37°. What is m∠1 so that m || n?


1 2 m

n

t

43

5 687

Suppose m∠3 5 5x 1 11 and m∠5 5 16x 1 1. What must the value of x be in order for line m to be parallel to line n?

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73. Make use of structure. Use the diagram below. Determine which pair of lines, if any, must be parallel for each statement to be true.

1 25 6

m

p

q9 1013 14

3 47 8

n

11 1215 16

a. ∠2 > ∠4

b. ∠2 is supplementary to ∠13.

c. ∠2 > ∠13

d. ∠5 > ∠10

e. ∠5 > ∠12

f. ∠2 > ∠4

74. Construct viable arguments. Complete the proof of the Converse of the Corresponding Angles Theorem.

DC

BA

P

E

F

Q

Given: ∠EPB > ∠EQD

Prove: � ��

AB CD

Statements Reasons

1. a. 1. Given2. b. 2. Definition of

congruent angles3. m∠EPB 1 m∠BPQ

5 1803. Linear Pair Postulate

4. m∠EQD 1 m∠BPQ 5 180

4. c.

5. ∠EQD and ∠BPQ are supplementary.

5. Definition of supplementary angles.

6. d. 6. Converse of Same-Side Interior Angles Postulate

75. A student found that m∠2 5 89° in the diagram shown. Which angle must have a measure of 91° in order for m and n to be parallel?

m

1 32 4

n

t

5 76 8

A. ∠3

B. ∠5

C. ∠6

D. ∠7

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LeSSon 7-3

76. In the diagram shown, ⊥AB PQ . If PT 5 TQ, which statement is true?

A

P

Q

B

T

A. AT 5 TB

B. PQ is the perpendicular bisector of AB.

C. AB is the perpendicular bisector of PQ .

D. PT is the perpendicular bisector of AB .

77. Reason quantitatively. Suppose p is the perpendicular bisector of RS in the diagram shown.

R p

T

S

a. If RT 5 5x 1 7 and RS 5 15x 2 1, what is the value of x?

b. If RT 5 7x 2 3 and TS 5 8, what is RS?

c. If RS 5 18 and TS 5 3x 2 1, what is the value of x?

d. Suppose � ��TQ forms a 35° angle with

� ��TR. What is

the measure of the angle formed by rays � ��TQ and ��

TS?

78. In the diagram shown, line m is parallel to line n, and point P is between lines m and n.

P

n

m

a. Determine the number of rays with endpoint P that are perpendicular to line n. Explain your answer.

b. Think about a ray with endpoint P that is perpendicular to line m. How is this ray related to the ray from Part a?


B

C

DQ

A

P

m n

Given: AB > BC

m is the ⊥ bisector of AB.

n is the ⊥ bisector of CD .

Prove: AP > DQ

Statements Reasons

1. AB > BC , m is the ⊥ bisector of AB, and n is the ⊥ bisector of CD.

1. Given

2. AB 5 CD 2. a.

3. AP AB12

53. Definition of

bisector

4. DQ CD12

54. b.

5. AP 5 DQ 5. Substitution6. c. 6. d.

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80. express regularity in repeated reasoning. In this diagram, AT 5 2x 1 3, CT 5 3x 2 1, BT 5 x 1 5, DT 5 4x 1 1, and m∠ATD 5 41x 1 8. If x 5 2, which segment is the perpendicular bisector of the other? Explain your reasoning.

A

T

B

C

D

LeSSon 8-1 81. Attend to precision. Use the ordered pairs A(3, 7),

B(22, 4), C(0, 5), and D(10, 0).

a. Find the slope of AB.

b. Find the slope of CD.

c. Find the slope of any line parallel to BC .

d. Find the slope of any line perpendicular to AD.

82. Use the three ordered pairs X(1, 0), Y(10, 3), and Z(15, 4). Which of the following statements CANNOT be true?

A. There is a line through Z that is parallel to XY� ��

.

B. There is a line through Z that is perpendicular to XY� ��

.

C. There is a line through Z that is the same line as XY� ��

.

D. There is a line through Z that intersects XY� ��

.

83. MN� ��

has a slope of 45

and PQ� ��

has a slope of 45

2 .

Are the lines parallel, perpendicular, or neither? Justify your answer.

84. � ��PQ contains the two points (0, 3) and (5, 27). The slope of

� ��RS is 1

2. Are the two lines parallel,

perpendicular, or neither? Justify your answer.

85. Make use of structure. For rectangle ABCD, two vertices are A(22, 3) and B(4, 6). Find the slopes of BC, CD, and DA. Explain your answer.

LeSSon 8-2 86. Consider the equation 2x 2 3y 5 18.

a. Write the equation in slope-intercept form.

b. Identify the slope and y-intercept of the line.

c. Another point on the line is (12, 2). Use that ordered pair to write an equation for the line in point-slope form.

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87. Use the two ordered pairs A(22, 9) and B(0, 1).

a. Suppose ⊥BC AB. Find the slope of � ��BC.

b. Write an equation in point-slope form for � ��BC.

c. Write an equation for � ��BC in slope-intercept

form.

d. Write an equation for � ��AB in point-slope form.

88. Which of the following is NOT an equation for a line perpendicular to y 5 2

3 x 2 1?

A. y x32

652 1

B. 3x 1 2y 5 5

C. 4y 5 26x

D. y x2 32

152 1

89. Reason abstractly. Suppose you are given two ordered pairs A and B. Explain how to write the equation of a line parallel to

� ��AB through a

given point.

90. Model with mathematics. A segment has endpoints P(24, 5) and Q(2, 21). Find the equation, in slope-intercept form, of the perpendicular bisector of PQ. Explain your solution.

Name class date Geometry Unit 1 Practice...H K L P e. T P S G 2.What is a correct name for this...

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