Transcript Chapter 4 Review

Homework
Lesson 4-1
∆𝑺𝑨𝑻 ≅ ∆𝑮𝑹𝑬. Complete each congruence statement
1. 𝑆 ≅ ∠𝑮
2. 𝐺𝑅 ≅ 𝑺𝑨
3. 𝐸 ≅ ∠𝑻
4. 𝐴𝑇 ≅ 𝑹𝑬
5. ∆𝐸𝑅𝐺 ≅ ∆𝑻𝑨𝑺
6. 𝐸𝐺 ≅ 𝑻𝑺
7. ∆𝑅𝐸𝐺 ≅ ∆𝑨𝑻𝑺
8. ∠𝑅 ≅ ∠𝑨
Mrs. Rivas
Homework
Mrs. Rivas
Lesson 4-1
State whether the figures are congruent. Justify your answers.
9. ∆𝐴𝐵𝐹; ∆𝐸𝐷𝐶
Yes; corresponding sides and corresponding angles are ≅.
Homework
Mrs. Rivas
Lesson 4-1
State whether the figures are congruent. Justify your answers.
10. ∆𝑇𝑈𝑉; ∆𝑈𝑉𝑊
No; the only corresponding part that is ≅ is 𝑼𝑽.
Homework
Mrs. Rivas
Lesson 4-1
State whether the figures are congruent. Justify your answers.
11. ⧠𝑋𝑌𝑍𝑉; ⧠ 𝑈𝑇𝑍𝑉
Yes; corresponding sides and corresponding angles are ≅.
Homework
Mrs. Rivas
Lesson 4-1
State whether the figures are congruent. Justify your answers.
12.∆𝐴𝐵𝐷; ∆𝐸𝐷𝐵
Yes; corresponding sides and corresponding angles are ≅.
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.
∠𝑻 ≅ ∠𝑺
𝑻𝒀 ≅ 𝑺𝑾
∠𝒀 ≅ ∠𝑾
𝑨𝑺𝑨
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..
𝑬𝑨 ≅ 𝑷𝑳
𝑨𝑳 ≅ 𝑳𝑨
∠𝑬𝑨𝑳 ≅ ∠𝑷𝑳𝑨
𝑺𝑨𝑺
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.
𝑵𝒐 𝒑𝒐𝒔𝒔𝒊𝒃𝒍𝒆
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.
𝑪𝑵 ≅ 𝑨𝑶
𝑵𝑨 ≅ 𝑶𝑪
𝑨𝑪 ≅ 𝑪𝑨
𝑺𝑺𝑺
Homework
Lessons 4-2 and 4-3
17. Given: 𝑃𝑋 ≅ 𝑃𝑌, 𝑍𝑃 bisects 𝑋𝑌 .
Prove: ∆𝑃𝑋𝑍  ∆𝑃𝑌𝑍
𝒁𝑷 bisects 𝑿𝒀 means that 𝑿𝑴 ≅ 𝒀𝑴.
𝑷𝑿 ≅ 𝑷𝒀 Given
𝑷𝒁 ≅ 𝑷𝒁 Reflective property of ≅
∆𝑷𝑿𝑴 ≅ ∆𝑷𝒀𝑴 SSS
∠𝑿𝑷𝑴 ≅ ∠𝒀𝑷𝑴 Def. of ≅.
𝑷𝒁 ≅ 𝑷𝒁 Reflective property of ≅
∆𝑷𝑿𝒁 ≅ ∆𝑷𝒀𝒁 by SAS.
Mrs. Rivas
Homework
Mrs. Rivas
Lessons 4-2 and 4-3
18. Given: ∠1 ≅ ∠2, ∠3 ≅ ∠4, 𝑃𝐷 ≅ 𝑃𝐶,
P is the midpoint of 𝐴𝐵.
Prove: ∆𝐴𝐷𝑃 ≅ ∆𝐵𝐶𝑃
It is given that 𝒎∠𝟏 ≅ 𝒎∠𝟐 and 𝒎∠𝟑 ≅ 𝒎∠𝟒.
By the ⦨ Addition. Post., 𝒎∠𝑫𝑷𝑨 ≅ 𝒎∠𝑪𝑷𝑩.
Since 𝑃 is the midpoint of 𝑨𝑩, 𝑷𝑨 ≅ 𝑷𝑩.
It is given that 𝑷𝑫 ≅ 𝑷𝑪, so ∆𝑨𝑫𝑷 ≅ ∆𝑩𝑪𝑷 by SAS.
Homework
Lesson 4-4
21. Given: 𝐿𝑂 ≅ 𝑀𝑁, 𝐿𝑂 ∥ 𝑀𝑁
Prove: ∠𝑀𝐿𝑁 ≅ ∠𝑂𝑁𝐿
𝑶𝑳 ∥ 𝑴𝑵, so ∠𝑶𝑳𝑵 ≅ ∠𝑴𝑵𝑳.
𝑳𝑵 ≅ 𝑳𝑵 by the Reflexive Prop. of ≅.
Since 𝑳𝑶 ≅ 𝑴𝑵, ∆𝑴𝑳𝑵 ≅ ∆𝑶𝑵𝑳 by SAS,
Then ∠𝑴𝑳𝑵 ≅ ∠𝑶𝑵𝑳 by CPCTC.
Mrs. Rivas
Homework
Lesson 4-4
22. Given: ∠𝑂𝑇𝑆 ≅ ∠𝑂𝐸𝑆, ∠𝐸𝑂𝑆 ≅ ∠𝑂𝑆𝑇
Prove: 𝑇𝑂 ≅ 𝐸𝑆
𝑶𝑺 ≅ 𝑶𝑺 Reflexive Prop. of ≅.
Since ∠𝑻 ≅ ∠𝑬 and ∠𝑻𝑺𝑶 ≅ ∠𝑬𝑶𝑺
∆𝑻𝑺𝑶 ≅ ∆𝑬𝑶𝑺 by AAS
𝑻𝑶 ≅ 𝑬𝑺 by CPCTC
Mrs. Rivas
Homework
Mrs. Rivas
Lesson 4-4
23. Given: ∠1 ≅ ∠2, ∠3 ≅ ∠4,
𝑀 is the midpoint of 𝑃𝑅
Prove: ∆𝑃𝑀𝑄 ≅ ∆𝑅𝑀𝑄
∠𝟏 ≅ ∠𝟐,∠𝟑 ≅ ∠𝟒 Given
𝑺𝑸 ≅ 𝑺𝑸 Reflexive Property
𝑴 is the midpoint of 𝑷𝑹, 𝒔𝒐 𝑷𝑴 ≅ 𝑹𝑴.
𝑸𝑴 ≅ 𝑸𝑴
Reflexive Property
SSS
∆𝑷𝐐𝐒 ≅ ∆𝑹𝑸𝑺
AAS
∆𝑷𝐐𝐌 ≅ ∆𝑹𝑸𝑴
𝑷𝑸 ≅ 𝑹𝑸
CPCTC
∠𝑷𝑸𝑴 ≅ ∠𝑹𝑴𝑸 CPCTC
Homework
Mrs. Rivas
Lesson 4-4
24. Given: 𝑃𝑂 = 𝑄𝑂, ∠1 ≅ ∠2,
Prove: ∠𝐴 ≅ ∠𝐵
∠𝟏 ≅ ∠𝟐,∠𝑨𝑷𝑶 ≅ ∠𝑩𝑸𝑶 Supplement of ≅ angles are ≅.
𝑷𝑶 ≅ 𝑸𝑶
Given
∠𝑷𝑶𝑨 ≅ ∠𝑩𝑶𝑸
Vertical ⦞ are ≅.
∆𝑶𝑷𝑸 ≅ ∆𝑩𝑸𝑶
ASA
∠𝑨 ≅ ∠𝑩
CPCTC
Homework
Mrs. Rivas
Lesson 4-5
Find the value of each variable
25.
𝟔𝟎
𝟔𝟎
𝟔𝟎
𝟐𝒙 + 𝟏𝟎 = 𝟔𝟎
𝟐𝒙 = 𝟓𝟎
𝒙 = 𝟐𝟓
Homework
Mrs. Rivas
Lesson 4-5
Find the value of each variable
26.
𝟐𝟓
𝟐𝟓 + 𝟗𝟎 + 𝒙 = 𝟏𝟖𝟎
𝟏𝟏𝟓 + 𝒙 = 𝟏𝟖𝟎
𝒙 = 𝟔𝟓
Homework
Mrs. Rivas
Lesson 4-5
Find the value of each variable
27.
𝒁 + 𝟏𝟑𝟓 = 𝟏𝟖𝟎
𝒛 = 𝟒𝟓
𝑰𝒔𝒐𝒔𝒄𝒆𝒍𝒆𝒔 𝑻𝒓𝒊𝒂𝒏𝒈𝒍𝒆𝒔 base angles
are congruent
𝒙 = 𝟒𝟓
𝑹𝒆𝒎𝒐𝒕𝒆 𝒂𝒏𝒈𝒍𝒆𝒔
𝒙 + 𝒚 = 𝟏𝟑𝟓
𝟒𝟓 + 𝒚 = 𝟏𝟑𝟓
𝒚 = 𝟗𝟎
Homework
Mrs. Rivas
Lesson 4-5
28. Given: ∠5 ≅ ∠6, 𝑃𝑋 ≅ 𝑃𝑌
Prove: ∆𝑃𝐴𝐵 is isosceles.
𝑷𝑿 ≅ 𝑷𝒀
Given
∠𝟏 ≅ ∠𝟐
Isosceles ∆ Theorem
∠𝟑 ≅ ∠𝟒
≅ Supplement Theorem
∠𝟓 ≅ ∠𝟔
Given
∆𝑷𝑿𝑨 ≅ ∆𝑷𝒀𝑩
𝑷𝑨 ≅ 𝑷𝑩
ASA
CPCTC
and ∆𝑷𝑨𝑩 is isosceles by Isosceles ∆ Theorem
Homework
Mrs. Rivas
Lesson 4-5
29. Given: 𝐴𝑃 ≅ 𝐵𝑃, 𝑃𝐶 ≅ 𝑃𝐷
Prove: ∆𝑄𝐶𝐷 is isosceles
𝑨𝑷 ≅ 𝑩𝑷 and 𝑷𝑪 ≅ 𝑷𝑫
Given
∠𝑨𝑷𝑪 ≅ ∠𝑩𝑷𝑫
Vertical ⦞ are ≅.
∆𝑨𝑷𝑪 ≅ ∆𝑩𝑷𝑫
SAS
∠𝑨𝑪𝑷 ≅ ∠𝑩𝑫𝑷
CPCTC
∠𝑷𝑪𝑫 ≅ ∠𝑷𝑫𝑪 Isosceles ∆ Theorem
𝒎∠𝑨𝑪𝑷 + 𝒎∠𝑷𝑪𝑫 = 𝒎∠𝑩𝑫𝑷 + 𝒎∠𝑷𝑫𝑪 Add. Prop. of ≅
so 𝒎∠𝑨𝑪𝑫 = 𝒎∠𝑩𝑫𝑪 by the ∠ Add. Post. and substitution.
𝑸𝑫 = 𝑸𝑪 by the Conv. of the Isosceles ∆ Theorem.
∆𝑸𝑪𝑫 is isosceles by Definition of Isosceles ∆.
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.
∆𝑨𝑹𝑶 ≅ ∆𝑹𝑨𝑭; 𝑯𝑳
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.
∆𝑹𝑸𝑴 ≅ ∆𝑸𝑹𝑺; 𝑺𝑺𝑺
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.
∆𝑨𝑶𝑵 ≅ ∆𝑴𝑶𝑷; 𝑨𝑨𝑺
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.
∆𝑮𝑹𝑻 ≅ ∆𝑮𝑺𝑶; 𝑨𝑺𝑨
Homework
Mrs. Rivas
Lessons 4-6 and 4-7
34. Given: M is the midpoint of 𝐴𝐵,
𝑀𝐶 ⊥ 𝐴𝐶, 𝑀𝐷 ⊥ 𝐵𝐷, ∠1 ≅ ∠2
Prove: ∆𝐴𝐶𝑀 ≅ ∆𝐵𝐷𝑀
∠𝟏 ≅ ∠𝟐 Given
𝑴𝑪 ≅ 𝑴𝑫 by the Conv. of the Isosceles ∆ Theorem.
𝑨𝑴 ≅ 𝑩𝑴 by the Def. of Midpoint
𝑴𝑪 ≅ 𝑨𝑪 means that ∠𝑨𝑪𝑴 is a right angle and ∆𝑨𝑪𝑴 is a right ∆.
𝑴𝑫 ⊥ 𝑩𝑫 means that ∠𝑩𝑫𝑴 is a right ⦨ and ∆𝑩𝑫𝑴 is a right ∆ .
∆𝑨𝑪𝑴 ≅ ∆𝑩𝑫𝑴 by HL.
Homework
Mrs. Rivas
35. The longest leg of ∆𝐴𝐵𝐶, 𝐴𝐵, measures 10 centimeters. 𝐵𝐶
measures 8 centimeters. You measure two of the legs of ∆𝑋𝑌𝑍 and find
that 𝐴𝐶 ≅ 𝑋𝑍 and 𝐵𝐶 ≅ 𝑌𝑍. 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 don’t know that there are right angles.