Practice B
Name———————————————————————— Lesson
2.7
Date —————————————
Practice B
For use with the lesson “Prove Angle Pair Relationships”
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
3
4
3. m∠ 1 1 m∠ 3 5 m∠ 2 1 m∠ 4. 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.
5. Adjacent complementary angles 6. Nonadjacent supplementary angles where one angle measures 428 where one angle measures 428
7. Congruent linear pairs
∠ ABC and ∠ CBD are adjacent 10. ∠ 1 and ∠ 2 are complementary. complementary angles. ∠ CBD ∠ 3 and ∠ 4 are complementary. and ∠ DBF are adjacent ∠ 1 and ∠ 3 are vertical angles. complementary angles.
Find the value of the variables and the measure of each angle in the diagram.
11.
Lesson 2.7
(13x 1 9)8
(4x 1 10)8
2(3y 2 25)8
13.
4y8
(17y 2 9)8 (5x 1 1)8
13x8
2(y 1 25)8
(4y 1 2)8 (15x 2 1)8
(21x 2 3)8
2-92
12.
(2y 2 30)8
14. 7x8 13y8 (5x 1 18)8 (16y 2 27)8
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
9.
8. Vertical angles which measure 428
Geometry Chapter Resource Book
CS10_CC_G_MECR710761_C2L07PB.indd 92
4/27/11 6:03:18 PM
Name———————————————————————— Lesson
2.7
Date —————————————
Practice B continued
For use with the lesson “Prove Angle Pair Relationships”
Give a reason for each step of the proof. 2
15. GIVEN: ∠ 2 > ∠ 3
PROVE: ∠ 1 > ∠ 4
Statements
3
1
4
Reasons
1. ∠ 2 > ∠ 3 1. ? 2. ∠ 3 > ∠ 4 2. ? 3. ∠ 2 > ∠ 4 3. ? 4. ∠ 1 > ∠ 2 4. ? 5. ∠ 1 > ∠ 4 5. ? 16. GIVEN: ∠ 1 and ∠ 2 are complementary.
1
∠ 1 > ∠ 3, ∠ 2 > ∠ 4
2
3
4
PROVE: ∠ 3 and ∠ 4 are complementary.
Statements
Reasons
2. m∠ 1 1 m∠ 2 5 908 2. ? 3. ∠ 1 > ∠ 3, ∠ 2 > ∠ 4 3. ? 4. m∠ 1 5 m∠ 3, m∠ 2 5 m∠ 4 4. ? 5. m∠ 3 1 m∠ 2 5 908 5. ? 6. m∠ 3 1 m∠ 4 5 908 6. ? 7. ∠ 3 and ∠ 4 are complementary. 7. ? 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 ? m∠ 1
1 3 4 5
2 6
7 9
Lesson 2.7
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
1. ∠ 1 and ∠ 2 are complementary. 1. ?
8
Geometry Chapter Resource Book
CS10_CC_G_MECR710761_C2L07PB.indd 93
2-93
4/27/11 6:03:18 PM
Lesson 2.6 Prove Statements about Segments and Angles, continued N
P
M
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
O
Statements Reasons } } > PQ , M is the 1. Given 1. NO } 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. NM 5 PM 8. Division Property of Equality } } > PM 9. Definition of 9. NM congruent segments
Lesson 2.7 Prove Angle Pair Relationships Teaching Guide
Check students’ circles. 1. ∠2 or ∠5 2. ∠2 and ∠5 are congruent. 3. ∠3 or ∠4 4. ∠3 and ∠4 are congruent. Practice Level A 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,
Practice Level B 1. false 2. true 3. false 4. true 5–10. Sample sketches are given. 5.
6. 428 428
1388 488
7.
8.
9.
A C
1388
428
428
1388
B
10.
D F
1 4
2
3
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. 1. Given 2. Definition of complementary angles 3. Given Geometry Chapter Resource Book
CS10_CC_G_MECR710761_C2AK.indd 29
answers
9.
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
A29
4/27/11 6:42:32 PM
4. Definition of congruent angles 5. Substitution Property of Equality 6. Substitution Property of Equality 7. Definition of complementary angles 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, 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∠ STU 5 908 by simplification and stating that ∠ STU is a right angle by definition. 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 supp 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)
Study Guide 1. 908 2. 908 3. 318 4. 1258 5. 528 6. 648 7. 908 8. 147 9. 44
A30
Problem Solving Workshop: Mixed Problem Solving 1. a. m∠ YWX 5 m∠ YWZ (Given)
m∠ XWZ 5 m∠ YWX 1 m∠ YWZ (Angle Add Post) m∠ XWZ 5 m∠ YWX 1 m∠ YWX (Subs Prop of Eq ) m∠ XWZ 5 2(m∠ YWX) (Simplify.) b. 288 2. a. H
H E
E S
S D
D G
G
C
C
b. Given: ES 5 GC, HE 5 SD 5 DG; Prove: HD 5 SC c.
Reasons Statements 1. ES 5 GC, 1. Given HE 5 SD 5 DG 2. HD 5 HE 1 ES 1 SD 2. Segment Addition Postulate 3. HD 5 DG 1 ES 1 SD 3. Substitution Prop. of Equality 4. HD 5 DG 1 GC 1 SD 4. Substitution Prop. of Equality 5. HD 5 SD 1 DG 1 GC 5. Commutative Prop. of Addition 6. HD 5 SC 6. Segment Addition Postulate 3. Sample answer: m∠ NMK 5 708, m∠ JML 5 1108 4. 68 5. 318; Substitute m∠ 3 for m∠ 2 in
the second equation. Then substitute 628 for m∠ 3 1 m∠ 4 in the first equation. Use this to find that m∠ 1 5 318. Then, m∠ 1 5 m∠ 2, so m∠ 2 5 318. Since m∠ 2 5 m∠ 3, m∠ 3 5 318. 6. a.
W 5 c(1 1 r) Given W 5 c 1 cr Distributive Property W 2 c 5 cr Subtraction Property of Equality W2c
Division Property of Equality b. 8% c. $12; Solve the original equation for c and then substitute for W and r to find c. d. Yes; Divide by c in the first step and then subtract 1. 5 r } c
Challenge Practice
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
answers
Lesson 2.7 Prove Angle Pair Relationships, continued
1. a. If m∠ 3 5 318, then m∠ 5 5 318. b. If m∠ 5 5 298, then m∠ 4 5 618.
Geometry Chapter Resource Book
CS10_CC_G_MECR710761_C2AK.indd 30
4/27/11 6:42:33 PM
2.7
Date —————————————
Practice B
For use with the lesson “Prove Angle Pair Relationships”
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
3
4
3. m∠ 1 1 m∠ 3 5 m∠ 2 1 m∠ 4. 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.
5. Adjacent complementary angles 6. Nonadjacent supplementary angles where one angle measures 428 where one angle measures 428
7. Congruent linear pairs
∠ ABC and ∠ CBD are adjacent 10. ∠ 1 and ∠ 2 are complementary. complementary angles. ∠ CBD ∠ 3 and ∠ 4 are complementary. and ∠ DBF are adjacent ∠ 1 and ∠ 3 are vertical angles. complementary angles.
Find the value of the variables and the measure of each angle in the diagram.
11.
Lesson 2.7
(13x 1 9)8
(4x 1 10)8
2(3y 2 25)8
13.
4y8
(17y 2 9)8 (5x 1 1)8
13x8
2(y 1 25)8
(4y 1 2)8 (15x 2 1)8
(21x 2 3)8
2-92
12.
(2y 2 30)8
14. 7x8 13y8 (5x 1 18)8 (16y 2 27)8
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
9.
8. Vertical angles which measure 428
Geometry Chapter Resource Book
CS10_CC_G_MECR710761_C2L07PB.indd 92
4/27/11 6:03:18 PM
Name———————————————————————— Lesson
2.7
Date —————————————
Practice B continued
For use with the lesson “Prove Angle Pair Relationships”
Give a reason for each step of the proof. 2
15. GIVEN: ∠ 2 > ∠ 3
PROVE: ∠ 1 > ∠ 4
Statements
3
1
4
Reasons
1. ∠ 2 > ∠ 3 1. ? 2. ∠ 3 > ∠ 4 2. ? 3. ∠ 2 > ∠ 4 3. ? 4. ∠ 1 > ∠ 2 4. ? 5. ∠ 1 > ∠ 4 5. ? 16. GIVEN: ∠ 1 and ∠ 2 are complementary.
1
∠ 1 > ∠ 3, ∠ 2 > ∠ 4
2
3
4
PROVE: ∠ 3 and ∠ 4 are complementary.
Statements
Reasons
2. m∠ 1 1 m∠ 2 5 908 2. ? 3. ∠ 1 > ∠ 3, ∠ 2 > ∠ 4 3. ? 4. m∠ 1 5 m∠ 3, m∠ 2 5 m∠ 4 4. ? 5. m∠ 3 1 m∠ 2 5 908 5. ? 6. m∠ 3 1 m∠ 4 5 908 6. ? 7. ∠ 3 and ∠ 4 are complementary. 7. ? 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 ? m∠ 1
1 3 4 5
2 6
7 9
Lesson 2.7
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
1. ∠ 1 and ∠ 2 are complementary. 1. ?
8
Geometry Chapter Resource Book
CS10_CC_G_MECR710761_C2L07PB.indd 93
2-93
4/27/11 6:03:18 PM
Lesson 2.6 Prove Statements about Segments and Angles, continued N
P
M
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
O
Statements Reasons } } > PQ , M is the 1. Given 1. NO } 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. NM 5 PM 8. Division Property of Equality } } > PM 9. Definition of 9. NM congruent segments
Lesson 2.7 Prove Angle Pair Relationships Teaching Guide
Check students’ circles. 1. ∠2 or ∠5 2. ∠2 and ∠5 are congruent. 3. ∠3 or ∠4 4. ∠3 and ∠4 are congruent. Practice Level A 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,
Practice Level B 1. false 2. true 3. false 4. true 5–10. Sample sketches are given. 5.
6. 428 428
1388 488
7.
8.
9.
A C
1388
428
428
1388
B
10.
D F
1 4
2
3
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. 1. Given 2. Definition of complementary angles 3. Given Geometry Chapter Resource Book
CS10_CC_G_MECR710761_C2AK.indd 29
answers
9.
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
A29
4/27/11 6:42:32 PM
4. Definition of congruent angles 5. Substitution Property of Equality 6. Substitution Property of Equality 7. Definition of complementary angles 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, 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∠ STU 5 908 by simplification and stating that ∠ STU is a right angle by definition. 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 supp 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)
Study Guide 1. 908 2. 908 3. 318 4. 1258 5. 528 6. 648 7. 908 8. 147 9. 44
A30
Problem Solving Workshop: Mixed Problem Solving 1. a. m∠ YWX 5 m∠ YWZ (Given)
m∠ XWZ 5 m∠ YWX 1 m∠ YWZ (Angle Add Post) m∠ XWZ 5 m∠ YWX 1 m∠ YWX (Subs Prop of Eq ) m∠ XWZ 5 2(m∠ YWX) (Simplify.) b. 288 2. a. H
H E
E S
S D
D G
G
C
C
b. Given: ES 5 GC, HE 5 SD 5 DG; Prove: HD 5 SC c.
Reasons Statements 1. ES 5 GC, 1. Given HE 5 SD 5 DG 2. HD 5 HE 1 ES 1 SD 2. Segment Addition Postulate 3. HD 5 DG 1 ES 1 SD 3. Substitution Prop. of Equality 4. HD 5 DG 1 GC 1 SD 4. Substitution Prop. of Equality 5. HD 5 SD 1 DG 1 GC 5. Commutative Prop. of Addition 6. HD 5 SC 6. Segment Addition Postulate 3. Sample answer: m∠ NMK 5 708, m∠ JML 5 1108 4. 68 5. 318; Substitute m∠ 3 for m∠ 2 in
the second equation. Then substitute 628 for m∠ 3 1 m∠ 4 in the first equation. Use this to find that m∠ 1 5 318. Then, m∠ 1 5 m∠ 2, so m∠ 2 5 318. Since m∠ 2 5 m∠ 3, m∠ 3 5 318. 6. a.
W 5 c(1 1 r) Given W 5 c 1 cr Distributive Property W 2 c 5 cr Subtraction Property of Equality W2c
Division Property of Equality b. 8% c. $12; Solve the original equation for c and then substitute for W and r to find c. d. Yes; Divide by c in the first step and then subtract 1. 5 r } c
Challenge Practice
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
answers
Lesson 2.7 Prove Angle Pair Relationships, continued
1. a. If m∠ 3 5 318, then m∠ 5 5 318. b. If m∠ 5 5 298, then m∠ 4 5 618.
Geometry Chapter Resource Book
CS10_CC_G_MECR710761_C2AK.indd 30
4/27/11 6:42:33 PM
