Download 2.7 Practice B

Name ———————————————————————
LESSON
2.7
Date ————————————
Practice B
For use with pages 123–133
Use the diagram to decide whether the statement is true or false.
1. If m∠ 1 5 478, then m∠ 2 5 438.
2. If m∠ 1 5 478, then m∠ 3 5 478.
2
1
4
3. m∠ 1 1 m∠ 3 5 m∠ 2 1 m∠ 4.
3
4. m∠ 1 1 m∠ 4 5 m∠ 2 1 m∠ 3.
Make a sketch of the given information. Label all angles which
can be determined.
6. Nonadjacent supplementary angles
where one angle measures 428
7. Congruent linear pairs
9. ∠ ABC and ∠ CBD are adjacent
complementary angles. ∠ CBD
and ∠ DBF are adjacent
complementary angles.
where one angle measures 428
8. Vertical angles which measure 428
10. ∠ 1 and ∠ 2 are complementary.
∠ 3 and ∠ 4 are complementary.
∠ 1 and ∠ 3 are vertical angles.
Find the value of the variables. Explain why your answer is reasonable.
11.
12.
(13x 1 9)8
(4x 1 10)8
2(3y 2 25)8
LESSON 2.7
(4y 1 2)8 (15x 2 1)8
140
13.
13x8
2(y 1 25)8
(2y 2 30)8
14.
4y8
(17y 2 9)8
(21x 2 3)8
Geometry
Chapter 2 Resource Book
(5x 1 1)8
7x8 13y8
(5x 1 18)8
(16y 2 27)8
Copyright © Holt McDougal. All rights reserved.
5. Adjacent complementary angles
Name ———————————————————————
LESSON
2.7
Practice B
For use with pages 123–133
Date ————————————
continued
Give a reason for each step of the proof.
2
15. GIVEN: ∠ 2 > ∠ 3
3
1
PROVE: ∠ 1 > ∠ 4
Statements
Reasons
1. ∠ 2 > ∠ 3
1.
?
2. ∠ 3 > ∠ 4
2.
?
3. ∠ 2 > ∠ 4
3.
?
4. ∠ 1 > ∠ 2
4.
?
5. ∠ 1 > ∠ 4
5.
?
4
16. Tell whether the proof is logically valid. If it is not, explain how to change the proof
so it is valid.
GIVEN: ∠ 1 and ∠ 2 are complementary.
1
∠ 1 > ∠ 3, ∠ 2 > ∠ 4
2
3
4
Statements
Reasons
1. ∠ 1 and ∠ 2 are complementary.
1. Given
2. m∠ 1 1 m∠ 2 5 908
2. Definition of complementary angles
3. ∠ 1 > ∠ 3, ∠ 2 > ∠ 4
3. Given
4. m∠ 3 1 m∠ 4 5 908
4. Substitution Property of Equality
5. ∠ 3 and ∠ 4 are complementary.
5. Definition of complementary angles
In the diagram, ∠ 1 is a right angle and m∠ 6 5 368. Complete the
statement with <, >, or 5.
17. m∠ 6 1 m∠ 7
?
m∠ 4 1 m∠ 5
18. m∠ 6 1 m∠ 8
?
m∠ 2 1 m∠ 3
19. m∠ 9
?
3(m∠ 6)
20. m∠ 2 1 m∠ 3
?
1
3
4 5
m∠ 1
2
6
7
9
LESSON 2.7
Copyright © Holt McDougal. All rights reserved.
PROVE: ∠ 3 and ∠ 4 are complementary.
8
Geometry
Chapter 2 Resource Book
141
Lesson 2.6, continued
Statements
}
}
9. NM > PM
8.
D
A
C
Lesson 2.7
B
3 4
G
Practice Level A
E
F
Statements
Reasons
###$ bisects ∠ ABC. 1. Given
1. BD
2. ∠ 1 > ∠ 2
2. Definition of angle
bisector
3. m∠ 1 5 m∠ 2
3. Definition of
congruent angles
4. m∠ 2 5 m∠ 3
4. Measures of vertical
angles are equal.
5. m∠ 1 5 m∠ 3
5. Transitive Property of
Equality
6. m∠ 1 5 m∠ 4
6. Measures of vertical
angles are equal.
7. m∠ 3 5 m∠ 4
7. Substitution Property
of Equality
###$ bisects ∠ EBG. 8. Definition of
8. BF
angle bisector
9.
N
1. ∠ A, ∠ B, ∠ C, and ∠ D are all congruent by
the Right Angles Congruence Theorem.
2. ∠ QRS, ∠ PVQ, and ∠ TVU are all congruent
by the Right Angles Congruence Theorem.
3. ∠ 1 > ∠ 3 by the Congruent Supplements
Theorem, because both angles are supplementary
to ∠ 2. 4. ∠ 1 > ∠ 3 by the Congruent
Complements Theorem, because both angles are
complementary to ∠ 2. 5. 658, 1158, 658
6. 1168, 1168, 648 7. 1128, 688, 688 8. 1138,
678, 1138 9. 44 10. 60 11. 14 12. 10 13. 13
14. 15 15. 388 16. 988 17. 1368
18. 448 19. 1428
20. The gap shows that the right angle of the
carpenter’s square is not congruent to the corner
of the door frame. The Right Angle Congruence
Theorem states that all right angles are
congruent, so the corner of the door frame is not
a right angle. 21. Given; ∠ 2; ∠ 4; Definition of
linear pair; ∠ 1 and ∠ 2 are supplementary; ∠ 3
and ∠ 4 are supplementary; Congruent
Supplements Theorem
M
Practice Level B
O
Statements
Reasons
}
}
1. NO > PQ, M is the 1. Given
}
midpoint of NO, M
is the midpoint
}
of PQ.
2. NO 5 PQ
2. Definition of
congruent segments
3. NM 5 MO,
3. Definition of
PM 5 MQ
midpoint
4. NO 5 NM 1 MO,
4. Segment Addition
PQ 5 PM 1 MQ
Postulate
5. NM 1 MO
5. Substitution Property
5 PM 1 MQ
of Equality
6. NM 1 NM
6. Substitution Property
5 PM 1 PM
of Equality
7. 2NM 5 2PM
7. Simplify.
8. Division Property of
8. NM 5 PM
Equality
A26
Geometry
Chapter 2 Resource Book
1. false 2. true 3. false 4. true
5–10. Sample sketches are given.
5.
6.
428
428
1388
488
7.
8.
9.
428
1388
A
C
1388
428
D
B
F
Copyright © Holt McDougal. All rights reserved.
ANSWERS
1 2
P
Reasons
9. Definition of
congruent segments
Lesson 2.7, continued
10.
1
4
2
3
Copyright © Holt McDougal. All rights reserved.
Statements
Reasons
1. ∠ 1 and ∠ 2 are
complementary.
1. Given
2. m∠ 1 1 m∠ 2 = 908
2. Definition of
complementary
angles
3. ∠ 1 > ∠ 3, ∠ 2 > ∠ 4
3. Given
4. m∠ 1 5 m∠ 3,
m∠ 2 5 m∠ 4
4. Definition
of congruent
angles
5. m∠ 3 1 m∠ 2 = 908
5. Substitution
Property of
Equality
6. m∠ 3 1 m∠ 4 = 908
7. ∠ 3 and ∠ 4 are
complementary.
6. Substitution
Property of
Equality
7. Definition of
complementary
angles
Review for Mastery
1. 908 2. 908 3. 318 4. 1258 5. 528 6. 648
7. 908 8. 147 9. 44
Problem Solving Workshop:
Mixed Problem Solving
1. a. m∠ YWX 5 m∠ YWZ (Given)
m∠ XWZ 5 m∠ YWX 1 m∠ YWZ (Angle
Addition Postulate)
m∠ XWZ 5 m∠ YWX 1 m∠ YWX
(Substitution Prop. of Eq.)
m∠ XWZ 5 2(m∠ YWX) (Simplify.)
17. 5 18. < 19. > 20. 5
Practice Level C
1. The Linear Pair Post. and Vertical Angles
Congruence Thm. can be used to deduce that ∠ 5,
∠ 6, and ∠ 7 are right angles. So, ∠ 5, ∠ 6, ∠ 7,
ANSWERS
11. x 5 5, y 5 26; Vertical angles are congruent
and 748 1 1068 5 1808
12. x 5 10, y 5 40; Vertical angles are congruent
and 508 1 1308 5 1808
13. x 5 7, y 5 9; Vertical angles are congruent
and 368 1 1448 5 1808
14. x 5 9, y 5 9; Vertical angles are congruent
and 638 1 1178 5 1808
15. 1. Given 2. Vertical angles are congruent.
3. Transitive Property of Congruence
4. Vertical angles are congruent. 5. Transitive
Property of Congruence 16. Not logically valid;
the Substitution Property of Equality cannot be
applied without first stating that m∠1 5 m∠3
and m∠2 5 m∠4 by the definition of congruent
angles. A complete valid proof is shown.
and ∠ 8 are all congruent by the Right Angles
Congruence Thm. ∠ 1 > ∠ 3 and ∠ 4 > ∠ 2 by
the Congruent Complements Thm.
2. By the Linear Pair Post., the following are
supplementary: ∠ 1 and ∠ 2, ∠ 3 and ∠ 4, ∠ 5
and ∠ 6, ∠ 7 and ∠ 9, ∠ 8 and ∠ 10. You can
deduce that ∠ 4 is a right angle, so ∠ 3 > ∠ 4 by
the Right Angles Congruence Thm. By the
Congruent Supplements Thm., ∠ 1, ∠ 6, ∠ 9, and
∠ 10 are congruent and ∠ 2, ∠ 5, ∠ 7, and ∠ 8
are congruent. 3. 378, 908, 538, 378 4. 568, 908,
568, 348 5. 518, 398, 908, 518 6. 548, 368, 368
7. x 5 25, y 5 14 8. x 5 13, y 5 16
9. x 5 50, y 5 53, z 5 127
10. x 5 4, y 5 21, z 5 71 11. 1188 12. 968
13. 848 14. 628 15. 288 16. 568 17. yes
18. no 19. no 20. no 21. yes 22. yes
23. Not logically valid; The Right Angles Congruence Theorem cannot be applied without first
stating that m∠ STU5 908 by simplification and
stating that ∠ STU is a right angle by the definition
of a right angle.
24. Sample answer:
1. ∠ 1 and ∠ 4 are comp. ∠ 4 and ∠ 5 are comp.
∠ 1 and ∠ 2 are supp. ∠ 5 and ∠ 6 are supp.
m∠ 1 5 528 (Given)
2. ∠ 1 > ∠ 5 (Congruent Complements Theorem)
3. ∠ 2 > ∠ 6 (Congruent Supplements Theorem)
4. m∠ 1 1 m∠ 2 5 1808
(Def. of supplementary angles)
5. 528 1 m∠ 2 5 1808 (Subst. Prop. of Equality)
6. m∠ 2 5 1288 (Subtraction Prop. of Equality)
7. m∠ 2 5 m∠ 6 (Def. of congruent angles)
8. m∠ 6 5 1288 (Subst. Prop. of Equality)
b. 288
Geometry
Chapter 2 Resource Book
A27