Chapter 4 Review

Download Report

Transcript Chapter 4 Review

Homework
Lesson 4-1
โˆ†๐‘บ๐‘จ๐‘ป โ‰… โˆ†๐‘ฎ๐‘น๐‘ฌ. Complete each congruence statement
1. ๏ƒ๐‘† โ‰… โˆ ๐‘ฎ
2. ๐บ๐‘… โ‰… ๐‘บ๐‘จ
3. ๏ƒ๐ธ โ‰… โˆ ๐‘ป
4. ๐ด๐‘‡ โ‰… ๐‘น๐‘ฌ
5. โˆ†๐ธ๐‘…๐บ โ‰… โˆ†๐‘ป๐‘จ๐‘บ
6. ๐ธ๐บ โ‰… ๐‘ป๐‘บ
7. โˆ†๐‘…๐ธ๐บ โ‰… โˆ†๐‘จ๐‘ป๐‘บ
8. โˆ ๐‘… โ‰… โˆ ๐‘จ
Mrs. Rivas
Homework
Mrs. Rivas
Lesson 4-1
State whether the figures are congruent. Justify your answers.
9. โˆ†๐ด๐ต๐น; โˆ†๐ธ๐ท๐ถ
Yes; corresponding sides and corresponding angles are โ‰….
Homework
Mrs. Rivas
Lesson 4-1
State whether the figures are congruent. Justify your answers.
10. โˆ†๐‘‡๐‘ˆ๐‘‰; โˆ†๐‘ˆ๐‘‰๐‘Š
No; the only corresponding part that is โ‰… is ๐‘ผ๐‘ฝ.
Homework
Mrs. Rivas
Lesson 4-1
State whether the figures are congruent. Justify your answers.
11. โง ๐‘‹๐‘Œ๐‘๐‘‰; โง  ๐‘ˆ๐‘‡๐‘๐‘‰
Yes; corresponding sides and corresponding angles are โ‰….
Homework
Mrs. Rivas
Lesson 4-1
State whether the figures are congruent. Justify your answers.
12.โˆ†๐ด๐ต๐ท; โˆ†๐ธ๐ท๐ต
Yes; corresponding sides and corresponding angles are โ‰….
Homework
Mrs. Rivas
Lessons 4-2 and 4-3
Can you prove the two triangles congruent? If so, write the congruence
statement and name the postulate you would use. If not, write not possible
and tell what other information you would need.
13.
โˆ ๐‘ป โ‰… โˆ ๐‘บ
๐‘ป๐’€ โ‰… ๐‘บ๐‘พ
โˆ ๐’€ โ‰… โˆ ๐‘พ
๐‘จ๐‘บ๐‘จ
Homework
Mrs. Rivas
Lessons 4-2 and 4-3
Can you prove the two triangles congruent? If so, write the congruence
statement and name the postulate you would use. If not, write not possible
and tell what other information you would need.
14..
๐‘ฌ๐‘จ โ‰… ๐‘ท๐‘ณ
๐‘จ๐‘ณ โ‰… ๐‘ณ๐‘จ
โˆ ๐‘ฌ๐‘จ๐‘ณ โ‰… โˆ ๐‘ท๐‘ณ๐‘จ
๐‘บ๐‘จ๐‘บ
Homework
Mrs. Rivas
Lessons 4-2 and 4-3
Can you prove the two triangles congruent? If so, write the congruence
statement and name the postulate you would use. If not, write not possible
and tell what other information you would need.
15.
๐‘ต๐’ ๐’‘๐’๐’”๐’”๐’Š๐’ƒ๐’๐’†
Homework
Mrs. Rivas
Lessons 4-2 and 4-3
Can you prove the two triangles congruent? If so, write the congruence
statement and name the postulate you would use. If not, write not possible
and tell what other information you would need.
16.
๐‘ช๐‘ต โ‰… ๐‘จ๐‘ถ
๐‘ต๐‘จ โ‰… ๐‘ถ๐‘ช
๐‘จ๐‘ช โ‰… ๐‘ช๐‘จ
๐‘บ๐‘บ๐‘บ
Homework
Lessons 4-2 and 4-3
17. Given: ๐‘ƒ๐‘‹ โ‰… ๐‘ƒ๐‘Œ, ๐‘๐‘ƒ bisects ๐‘‹๐‘Œ .
Prove: โˆ†๐‘ƒ๐‘‹๐‘ ๏€ โˆ†๐‘ƒ๐‘Œ๐‘
๐’๐‘ท bisects ๐‘ฟ๐’€ means that ๐‘ฟ๐‘ด โ‰… ๐’€๐‘ด.
๐‘ท๐‘ฟ โ‰… ๐‘ท๐’€ Given
๐‘ท๐’ โ‰… ๐‘ท๐’ Reflective property of โ‰…
โˆ†๐‘ท๐‘ฟ๐‘ด โ‰… โˆ†๐‘ท๐’€๐‘ด SSS
โˆ ๐‘ฟ๐‘ท๐‘ด โ‰… โˆ ๐’€๐‘ท๐‘ด Def. of โ‰….
๐‘ท๐’ โ‰… ๐‘ท๐’ Reflective property of โ‰…
โˆ†๐‘ท๐‘ฟ๐’ โ‰… โˆ†๐‘ท๐’€๐’ by SAS.
Mrs. Rivas
Homework
Mrs. Rivas
Lessons 4-2 and 4-3
18. Given: โˆ 1 โ‰… โˆ 2, โˆ 3 โ‰… โˆ 4, ๐‘ƒ๐ท โ‰… ๐‘ƒ๐ถ,
P is the midpoint of ๐ด๐ต.
Prove: โˆ†๐ด๐ท๐‘ƒ โ‰… โˆ†๐ต๐ถ๐‘ƒ
It is given that ๐’Žโˆ ๐Ÿ โ‰… ๐’Žโˆ ๐Ÿ and ๐’Žโˆ ๐Ÿ‘ โ‰… ๐’Žโˆ ๐Ÿ’.
By the โฆจ Addition. Post., ๐’Žโˆ ๐‘ซ๐‘ท๐‘จ โ‰… ๐’Žโˆ ๐‘ช๐‘ท๐‘ฉ.
Since ๐‘ƒ is the midpoint of ๐‘จ๐‘ฉ, ๐‘ท๐‘จ โ‰… ๐‘ท๐‘ฉ.
It is given that ๐‘ท๐‘ซ โ‰… ๐‘ท๐‘ช, so โˆ†๐‘จ๐‘ซ๐‘ท โ‰… โˆ†๐‘ฉ๐‘ช๐‘ท by SAS.
Homework
Lesson 4-4
21. Given: ๐ฟ๐‘‚ โ‰… ๐‘€๐‘, ๐ฟ๐‘‚ โˆฅ ๐‘€๐‘
Prove: โˆ ๐‘€๐ฟ๐‘ โ‰… โˆ ๐‘‚๐‘๐ฟ
๐‘ถ๐‘ณ โˆฅ ๐‘ด๐‘ต, so โˆ ๐‘ถ๐‘ณ๐‘ต โ‰… โˆ ๐‘ด๐‘ต๐‘ณ.
๐‘ณ๐‘ต โ‰… ๐‘ณ๐‘ต by the Re๏ฌ‚exive Prop. of โ‰….
Since ๐‘ณ๐‘ถ โ‰… ๐‘ด๐‘ต, โˆ†๐‘ด๐‘ณ๐‘ต โ‰… โˆ†๐‘ถ๐‘ต๐‘ณ by SAS,
Then โˆ ๐‘ด๐‘ณ๐‘ต โ‰… โˆ ๐‘ถ๐‘ต๐‘ณ by CPCTC.
Mrs. Rivas
Homework
Lesson 4-4
22. Given: โˆ ๐‘‚๐‘‡๐‘† โ‰… โˆ ๐‘‚๐ธ๐‘†, โˆ ๐ธ๐‘‚๐‘† โ‰… โˆ ๐‘‚๐‘†๐‘‡
Prove: ๐‘‡๐‘‚ โ‰… ๐ธ๐‘†
๐‘ถ๐‘บ โ‰… ๐‘ถ๐‘บ Re๏ฌ‚exive Prop. of โ‰….
Since โˆ ๐‘ป โ‰… โˆ ๐‘ฌ and โˆ ๐‘ป๐‘บ๐‘ถ โ‰… โˆ ๐‘ฌ๐‘ถ๐‘บ
โˆ†๐‘ป๐‘บ๐‘ถ โ‰… โˆ†๐‘ฌ๐‘ถ๐‘บ by AAS
๐‘ป๐‘ถ โ‰… ๐‘ฌ๐‘บ by CPCTC
Mrs. Rivas
Homework
Mrs. Rivas
Lesson 4-4
23. Given: โˆ 1 โ‰… โˆ 2, โˆ 3 โ‰… โˆ 4,
๐‘€ is the midpoint of ๐‘ƒ๐‘…
Prove: โˆ†๐‘ƒ๐‘€๐‘„ โ‰… โˆ†๐‘…๐‘€๐‘„
โˆ ๐Ÿ โ‰… โˆ ๐Ÿ,โˆ ๐Ÿ‘ โ‰… โˆ ๐Ÿ’ Given
๐‘บ๐‘ธ โ‰… ๐‘บ๐‘ธ Reflexive Property
๐‘ด is the midpoint of ๐‘ท๐‘น, ๐’”๐’ ๐‘ท๐‘ด โ‰… ๐‘น๐‘ด.
๐‘ธ๐‘ด โ‰… ๐‘ธ๐‘ด
Reflexive Property
SSS
โˆ†๐‘ท๐๐’ โ‰… โˆ†๐‘น๐‘ธ๐‘บ
AAS
โˆ†๐‘ท๐๐Œ โ‰… โˆ†๐‘น๐‘ธ๐‘ด
๐‘ท๐‘ธ โ‰… ๐‘น๐‘ธ
CPCTC
โˆ ๐‘ท๐‘ธ๐‘ด โ‰… โˆ ๐‘น๐‘ด๐‘ธ CPCTC
Homework
Mrs. Rivas
Lesson 4-4
24. Given: ๐‘ƒ๐‘‚ = ๐‘„๐‘‚, โˆ 1 โ‰… โˆ 2,
Prove: โˆ ๐ด โ‰… โˆ ๐ต
โˆ ๐Ÿ โ‰… โˆ ๐Ÿ,โˆ ๐‘จ๐‘ท๐‘ถ โ‰… โˆ ๐‘ฉ๐‘ธ๐‘ถ Supplement of โ‰… angles are โ‰….
๐‘ท๐‘ถ โ‰… ๐‘ธ๐‘ถ
Given
โˆ ๐‘ท๐‘ถ๐‘จ โ‰… โˆ ๐‘ฉ๐‘ถ๐‘ธ
Vertical โฆž are โ‰….
โˆ†๐‘ถ๐‘ท๐‘ธ โ‰… โˆ†๐‘ฉ๐‘ธ๐‘ถ
ASA
โˆ ๐‘จ โ‰… โˆ ๐‘ฉ
CPCTC
Homework
Mrs. Rivas
Lesson 4-5
Find the value of each variable
25.
๐Ÿ”๐ŸŽ
๐Ÿ”๐ŸŽ
๐Ÿ”๐ŸŽ
๐Ÿ๐’™ + ๐Ÿ๐ŸŽ = ๐Ÿ”๐ŸŽ
๐Ÿ๐’™ = ๐Ÿ“๐ŸŽ
๐’™ = ๐Ÿ๐Ÿ“
Homework
Mrs. Rivas
Lesson 4-5
Find the value of each variable
26.
๐Ÿ๐Ÿ“
๐Ÿ๐Ÿ“ + ๐Ÿ—๐ŸŽ + ๐’™ = ๐Ÿ๐Ÿ–๐ŸŽ
๐Ÿ๐Ÿ๐Ÿ“ + ๐’™ = ๐Ÿ๐Ÿ–๐ŸŽ
๐’™ = ๐Ÿ”๐Ÿ“
Homework
Mrs. Rivas
Lesson 4-5
Find the value of each variable
27.
๐’ + ๐Ÿ๐Ÿ‘๐Ÿ“ = ๐Ÿ๐Ÿ–๐ŸŽ
๐’› = ๐Ÿ’๐Ÿ“
๐‘ฐ๐’”๐’๐’”๐’„๐’†๐’๐’†๐’” ๐‘ป๐’“๐’Š๐’‚๐’๐’ˆ๐’๐’†๐’” base angles
are congruent
๐’™ = ๐Ÿ’๐Ÿ“
๐‘น๐’†๐’Ž๐’๐’•๐’† ๐’‚๐’๐’ˆ๐’๐’†๐’”
๐’™ + ๐’š = ๐Ÿ๐Ÿ‘๐Ÿ“
๐Ÿ’๐Ÿ“ + ๐’š = ๐Ÿ๐Ÿ‘๐Ÿ“
๐’š = ๐Ÿ—๐ŸŽ
Homework
Mrs. Rivas
Lesson 4-5
28. Given: โˆ 5 โ‰… โˆ 6, ๐‘ƒ๐‘‹ โ‰… ๐‘ƒ๐‘Œ
Prove: โˆ†๐‘ƒ๐ด๐ต is isosceles.
๐‘ท๐‘ฟ โ‰… ๐‘ท๐’€
Given
โˆ ๐Ÿ โ‰… โˆ ๐Ÿ
Isosceles โˆ† Theorem
โˆ ๐Ÿ‘ โ‰… โˆ ๐Ÿ’
โ‰… Supplement Theorem
โˆ ๐Ÿ“ โ‰… โˆ ๐Ÿ”
Given
โˆ†๐‘ท๐‘ฟ๐‘จ โ‰… โˆ†๐‘ท๐’€๐‘ฉ
๐‘ท๐‘จ โ‰… ๐‘ท๐‘ฉ
ASA
CPCTC
and โˆ†๐‘ท๐‘จ๐‘ฉ is isosceles by Isosceles โˆ† Theorem
Homework
Mrs. Rivas
Lesson 4-5
29. Given: ๐ด๐‘ƒ โ‰… ๐ต๐‘ƒ, ๐‘ƒ๐ถ โ‰… ๐‘ƒ๐ท
Prove: โˆ†๐‘„๐ถ๐ท is isosceles
๐‘จ๐‘ท โ‰… ๐‘ฉ๐‘ท and ๐‘ท๐‘ช โ‰… ๐‘ท๐‘ซ
Given
โˆ ๐‘จ๐‘ท๐‘ช โ‰… โˆ ๐‘ฉ๐‘ท๐‘ซ
Vertical โฆž are โ‰….
โˆ†๐‘จ๐‘ท๐‘ช โ‰… โˆ†๐‘ฉ๐‘ท๐‘ซ
SAS
โˆ ๐‘จ๐‘ช๐‘ท โ‰… โˆ ๐‘ฉ๐‘ซ๐‘ท
CPCTC
โˆ ๐‘ท๐‘ช๐‘ซ โ‰… โˆ ๐‘ท๐‘ซ๐‘ช Isosceles โˆ† Theorem
๐’Žโˆ ๐‘จ๐‘ช๐‘ท + ๐’Žโˆ ๐‘ท๐‘ช๐‘ซ = ๐’Žโˆ ๐‘ฉ๐‘ซ๐‘ท + ๐’Žโˆ ๐‘ท๐‘ซ๐‘ช Add. Prop. of โ‰…
so ๐’Žโˆ ๐‘จ๐‘ช๐‘ซ = ๐’Žโˆ ๐‘ฉ๐‘ซ๐‘ช by the โˆ  Add. Post. and substitution.
๐‘ธ๐‘ซ = ๐‘ธ๐‘ช by the Conv. of the Isosceles โˆ† Theorem.
โˆ†๐‘ธ๐‘ช๐‘ซ is isosceles by Definition of Isosceles โˆ†.
Homework
Mrs. Rivas
Lessons 4-6 and 4-7
Name a pair of overlapping congruent triangles in each diagram. State
whether the triangles are congruent by SSS, SAS, ASA, AAS, or HL.
30.
โˆ†๐‘จ๐‘น๐‘ถ โ‰… โˆ†๐‘น๐‘จ๐‘ญ; ๐‘ฏ๐‘ณ
Homework
Mrs. Rivas
Lessons 4-6 and 4-7
Name a pair of overlapping congruent triangles in each diagram. State
whether the triangles are congruent by SSS, SAS, ASA, AAS, or HL.
31.
โˆ†๐‘น๐‘ธ๐‘ด โ‰… โˆ†๐‘ธ๐‘น๐‘บ; ๐‘บ๐‘บ๐‘บ
Homework
Mrs. Rivas
Lessons 4-6 and 4-7
Name a pair of overlapping congruent triangles in each diagram. State
whether the triangles are congruent by SSS, SAS, ASA, AAS, or HL.
32.
โˆ†๐‘จ๐‘ถ๐‘ต โ‰… โˆ†๐‘ด๐‘ถ๐‘ท; ๐‘จ๐‘จ๐‘บ
Homework
Mrs. Rivas
Lessons 4-6 and 4-7
Name a pair of overlapping congruent triangles in each diagram. State
whether the triangles are congruent by SSS, SAS, ASA, AAS, or HL.
33.
โˆ†๐‘ฎ๐‘น๐‘ป โ‰… โˆ†๐‘ฎ๐‘บ๐‘ถ; ๐‘จ๐‘บ๐‘จ
Homework
Mrs. Rivas
Lessons 4-6 and 4-7
34. Given: M is the midpoint of ๐ด๐ต,
๐‘€๐ถ โŠฅ ๐ด๐ถ, ๐‘€๐ท โŠฅ ๐ต๐ท, โˆ 1 โ‰… โˆ 2
Prove: โˆ†๐ด๐ถ๐‘€ โ‰… โˆ†๐ต๐ท๐‘€
โˆ ๐Ÿ โ‰… โˆ ๐Ÿ Given
๐‘ด๐‘ช โ‰… ๐‘ด๐‘ซ by the Conv. of the Isosceles โˆ† Theorem.
๐‘จ๐‘ด โ‰… ๐‘ฉ๐‘ด by the Def. of Midpoint
๐‘ด๐‘ช โ‰… ๐‘จ๐‘ช means that โˆ ๐‘จ๐‘ช๐‘ด is a right angle and โˆ†๐‘จ๐‘ช๐‘ด is a right โˆ†.
๐‘ด๐‘ซ โŠฅ ๐‘ฉ๐‘ซ means that โˆ ๐‘ฉ๐‘ซ๐‘ด is a right โฆจ and โˆ†๐‘ฉ๐‘ซ๐‘ด is a right โˆ† .
โˆ†๐‘จ๐‘ช๐‘ด โ‰… โˆ†๐‘ฉ๐‘ซ๐‘ด by HL.
Homework
Mrs. Rivas
35. The longest leg of โˆ†๐ด๐ต๐ถ, ๐ด๐ต, measures 10 centimeters. ๐ต๐ถ
measures 8 centimeters. You measure two of the legs of โˆ†๐‘‹๐‘Œ๐‘ and find
that ๐ด๐ถ โ‰… ๐‘‹๐‘ and ๐ต๐ถ โ‰… ๐‘Œ๐‘. Can you conclude that two triangles to be
congruent by the HL Theorem? Explain why or why not.
No; you only know that two sides (SS) are congruent, and
you donโ€™t know that there are right angles.