Download Tools of Geometry

Name _______________________________________ Date ___________________ Class __________________
Tools of Geometry
Module 1
Choose the best answer.
Refer to the figure for Exercises 1 and 2.
1. Which represents the name of the ray
whose endpoint is K and that passes
through R?
A RK
C KS
B KT
D RK
2. In the diagram, how many different rays
have endpoint R?
F 1
H 3
G 2
J 4
8. What is the next letter in the sequence?
D, H, L, P, . . .
F Q
H S
G R
J T
9. Which is the counterexample that proves
the conjecture false?
“If two rays have the same endpoint, then
they are opposite rays.”
A
C
B
D
Refer to the figure for Exercises 3 and 4.
3. What is MP?
A 1
C 4
B 2
D 5
4. What is LP?
F 7.5
H 2.5
G 2.5
J 7.5
5. An angle whose measure is 70° is what
type of angle?
A acute
C obtuse
B right
D straight
6. GJ bisects FGH, mFGJ  (7x  9)°,
and mHGJ  (2x  36)°. What is
mFGH?
F 43°
H 86°
G 54°
J 108°
7. An angle measuring 22° is bisected.
What is the measure of the angles that
are formed?
A 11°
C 33°
B 22°
D 44°
10. Identify the hypothesis of the conditional
statement “Two angles are
complementary if the sum of their
measures is 90 degrees.”
F if
G Two angles are complementary
H the sum of their measures is
90 degrees
J Two angles are complementary if the
sum of their measures is 90 degrees.
11. Which conditional statement has the
same truth value as this statement?
“The sum of two odd numbers is even.”
A If two even numbers are added, then
their sum is even.
B If an even and odd number are
added, then their sum is even.
C If two even numbers are multiplied,
then their product is odd.
D If two odd numbers are multiplied,
then their product is even.
© Houghton Mifflin Harcourt Publishing Company
Holt McDougal Analytic Geometry
Name _______________________________________ Date ___________________ Class __________________
Tools of Geometry
Use the figure for Exercises 12–15.
18. Find the next item in the pattern.
2, 5, 8, 11, 14, . . .
________________________________________
12. Name a line.
19. Show that the conjecture is false by
finding a counterexample. When the
letters i and e appear next to each other
in a word, the letter i always comes
before the letter e.
____________________________________
________________________________________
13. Name a segment on line n.
____________________________________
14. Name a ray with endpoint A.
20. Identify the hypothesis and conclusion
of the statement “If it is raining, then
there are clouds in the sky.”
________________________________________
____________________________________
________________________________________
15. Name the intersection of BC and AB.
________________________________________
____________________________________
16. Z is in the interior of WXY.
If mWXZ  110°, and mZXY  20°,
what is mWXY?
21. Given: If Lewis earns a scholarship, he
can go to college. Lewis earns a
scholarship.
Conjecture: Lewis can go to college.
Determine whether the conjecture is
valid by the Law of Detachment.
____________________________________
17. A and B are complementary.
mA  29. Find mB.
________________________________________
____________________________________
© Houghton Mifflin Harcourt Publishing Company
Holt McDougal Analytic Geometry
Name _______________________________________ Date ___________________ Class __________________
Algebraic and Geometric Proofs
Module 2
Choose the best answer.
1. Consider the related biconditional
statement for the conditional statement “If
Shelly lives in Texas, then she lives in the
United States.”
Which of the following statements is true
about the related biconditional
statement?
A The biconditional is true because the
conditional is true.
B The biconditional is false because
the conditional and its converse are
false.
C The biconditional is true because the
conditional and its converse are true.
D The biconditional is false because
the converse of the conditional is
false.
2. If r  14  9, what justifies r  23?
F Transitive Property of Equality
G Subtraction Property of Equality
4. If 5  2k, what justifies 2k  5?
F Multiplication Property of Equality
G Division Property of Equality
H Symmetric Property of Equality
J Reflexive Property of Equality
5. Which completes the statement?
If 6x  5 and d  6x, then ______ by
the Transitive Property of Equality.
A d5
B x
5
6
C 6x  d
D x
d
6
6. Which completes the statement?
If RS  GH, then ______ by the
Symmetric Property of Congruence.
F RS  GH
H GH  RS
G RS  RS
J RS  GH
7. Given: L bisects KM ; M bisects LN.
Prove: KL  MN
H Symmetric Property of Equality
J Reflexive Property of Equality
x 1
 8, what justifies
2
x  1  16?
3. If
A Subtraction Property of Equality
B Division Property of Equality
Proof:
Since L bisects KM and M bisects LN,
by definition of bisect, KL  LM and
LM  MN. Then, by the
?
,
KL  MN. Finally,
KL  MN by the definition of congruent
segments.Which completes the proof?
C Transitive Property of Equality
A Common Segments Theorem
D Multiplication Property of Equality
B Transitive Property of Congruence
C Segment Addition Postulate
D Symmetric Property of Congruence
© Houghton Mifflin Harcourt Publishing Company
Holt McDougal Analytic Geometry
Name _______________________________________ Date ___________________ Class __________________
Algebraic and Geometric Proofs
8. Given: 1  4
Prove: 2  3
Proof:
11. Complete the sentence “A _________
is any statement that you can prove.”
Statements
Reasons
1. 1  4
1. Given
2. 1 and 2 are supp.,
and 3 and 4 are
supp.
2. Lin. Pairs
Thm.
3. 2  3
3.
________________________________________
12. What is the reason for Step 2?
Statements
Reasons
1. 1  2 and
2  3.
1. Given
2. m1  m2 and
m2  m3.
2.
F Transitive Property of Congruence
3. m1  m3
3. Trans. Prop. of 
G Vertical Angles Theorem
4. 1  3
4. Def. of  s
?
Which completes the proof?
H Congruent Supplements Theorem
J Angle Addition Postulate
9. For the following statement, what is the
“Prove” statement?
If ABCD is a rhombus, then it is a
parallelogram.
?
________________________________________
________________________________________
13. The box is part of a flowchart proof.
Identify the statement.
A ABCD is a quadrilateral.
B ABCD is a rhombus.
C ABCD is a parallelogram.
D ABCD is not a square.
10. Use the Reflexive Property of
Congruence to complete the statement
“A  ________.”
________________________________________
________________________________________
14. Write True or False. A paragraph proof
is less formal than a two-column proof,
so you do not need to include every
step.
________________________________________
© Houghton Mifflin Harcourt Publishing Company
Holt McDougal Analytic Geometry
Name _______________________________________ Date ___________________ Class __________________
Proving Theorems about Lines and Angles
Module 3
Choose the best answer.
6. What is the value of x?
Refer to the figure for Exercises 1–3.
1. Which segment is perpendicular to DE ?
A AB
C DF
B CF
D EF
F 35
H 15
G 20
J 12.5
7. Which could you use to show that u || v?
2. Which segment is parallel to BE ?
F AB
H CF
G BC
J DF
3. Which segment is NOT skew to DF ?
A 1 and 8 are supplementary.
B 4 and 8 are supplementary.
A AB
C BC
C 3 and 7 are congruent.
B AC
D BE
D 7 and 8 are congruent.
Refer to the figure for Exercises 4–5.
8. Why is m  n?
F If two coplanar lines are
perpendicular to the same line, then
the two lines are parallel to each
other.
4. Which pair of angles are corresponding
angles?
F 1 and 4
H 2 and 5
G 6 and 8
J 8 and 7
5. Which pair of angles are alternate
exterior angles?
A 4 and 8
C 3 and 7
B 2 and 5
D 1 and 4
G If two parallel lines are cut by a
transversal, then the pairs of sameside interior angles are
supplementary.
H In a plane, if a transversal is
perpendicular to one of two parallel
lines, then it is perpendicular to the
other line.
J If two intersecting lines form a linear
pair of congruent angles, then the
lines are perpendicular.
Proving Theorems about Lines and Angles
9. Identify a pair of parallel segments.
________________________________________
10. Write True or False. Parallel lines
intersect.
________________________________________
11. How many angles are formed by two
lines and a transversal?
________________________________________
15. Given r  s. What is the measure of 1?
________________________________________
16. Write True or False. You can use the
measures of the angles formed by two
lines and a transversal to determine
whether the two lines are parallel.
________________________________________
17. If 2  8, then r  s by which
theorem?
12. What is the name given to the angle
pair 3 and 5?
________________________________________
________________________________________
13. If parallel lines are intersected by a
transversal, how many pairs of
corresponding angles are there?
________________________________________
14. Complete the sentence. If a transversal
intersects parallel lines and an obtuse
angle is formed, all the obtuse angles
are ________.
18. If two coplanar lines are cut by a
transversal so that right angles are
formed, how many different angle
measures are there?
________________________________________
19. Name the shortest segment from
C to AB.
________________________________________
________________________________________
Congruence and Triangles
Module 4
Choose the best answer.
1. The polygon Q(3, 2), R(6, 5), S(6, 2)
undergoes the transformation:
(x, y)  (4x, 4y). Name the coordinates
of the image points.
A Q'(3, 2), R'(6, 5), S'(6, 2)
B Q'(7, 6), R'(10, 9), S'(10, 6)
7. What is the value of x if the acute angles
of a right triangle measure 8x and 12x?
A 4.5
C 9
B 5
D 10
8. In the figure, PQR  UVW. What is
mR?
C Q'(12, 8), R'(24, 20), S'(24, 8)
D Q'(–1, –2), R'(2, 1), S'(2, –2)
2. The angles of a triangle measure 4,
86, and 90. Which classification of
the triangle is correct?
F acute
H obtuse
G equiangular
J right
3. Which could be the angle measures
of an acute triangle?
A 32-56-92
C 34-56-90
B 33-58-91
D 35-56-89
4. What is the length of the longest side
of the triangle?
F 8.5
H 19
G 26
J 40
F 42
H 88
G 72
J 92
9. If KMQ  WJR, which segment is
congruent to RW ?
A KM
C QK
B MQ
D JW
10. Which angle is congruent to Z if
ZLV  SPN ?
F V
H N
G S
J P
11. If ABC  KJC, which statement is NOT
necessarily true?
5. What is mW ?
A 35
C 70
A ACB  KCJ C JC  BC
B 40.3
D 81
B J  B
6. Two angles of a triangle measure 22 and
53. What is the measure of the
third angle?
D JK  BC
12. Suppose S  N, J  I,
A  T, JS  IN, JA  IT , and
AS  TN. Which is true?
F 15
H 75
F JSA  ITN
H JSA  INT
G 25
J 105
G JSA  NTI
J JSA  TNI
Congruence and Triangles
13. The transformation M:
(x, y)  (0.5x, 0.5y) has been applied
to the polygon KLM. If the point M had
coordinates (4, 4), name the
coordinates of the image of M.
________________________________________
20. If KLM  RST, what is the value
of x?
________________________________________
14. Is the polygon K(5, 4), L(5, 6), M(7, 6)
congruent to the polygon N(4, 3), P(4,
5), Q(6, 5)?
________________________________________
Use the figure for Exercises 15 and
16.
________________________________________
21. Complete the statement. Two triangles
are congruent if and only if their
_______ angles and sides are
congruent.
________________________________________
Use the figure for Exercises 22 and 23.
15. Classify the triangle by its angle
measures.
________________________________________
22. What value of x proves ABC  DEF
by SAS?
16. Classify the triangle by its side lengths.
________________________________________
17. Complete the sentence. All of the angles
in an equilateral triangle measure
_________.
________________________________________
18. What is the measure of 1?
_________________________________
23. If AB  DE , what additional
congruence statement is needed to
prove ABC  DEF by SSS?
_________________________________
Use the figure for Exercises 24 and 25.
24. Write True or False. You can use AAS
to prove ABE  CDE.
________________________________________
19. Given: GHJ  NOP. What is the
value of x?
________________________________________
25. What additional congruence statement
is needed to prove ABE  CDE by
HL?
Module 1 Quiz
1. C
2. G
3. C
4. J
5. A
6. J
7. A
8. J
9. C
10 H.
11. A
12. Possible answers: BC, CB, or
line n
13. BC or CB
14. AD
15. point B
16. 130°
17. 61°
18. 17
19. Possible answer: receive
20. Hypothesis: It is raining.
Conclusion: There are clouds in
the sky.
21. valid
Module 2 Quiz
1. D
2. G
3. D
4. H
5. A
6. H
7. B
8. H
9. C
10. ÐA
11. theorem
12. Def. of  s
13. mA  mB  180
14. False
Module 3 Quiz
1. D
2. H
3. B
4. G
5. A
6. G
7. C
8. J
9. Sample answer: AB and DC
10. False
11. eight
12. alternate interior angles
13. four
14. congruent
15. 135°
16. True
17. Conv. of the Alt. Ext. s Thm.
18. one
19. CA
Module 4 Quiz
1. C
2. J
3. D
4. J
5. D
6. J
7. A
8. G
9. C
10. G
11. D
12. H
13. M9(2,2)
14.yes
15. acute
16. isosceles
17. 60°
18. 45°
19. 35
20. 5
21. corresponding
22. 6
23. AC  DF
24. True
25. EC  EA