Download #1)

Similar Triangle Proofs (1) HW ANSWER KEY
#1)
Statement
Reason
̅̅̅̅
̅̅̅̅
1)
Given
1) 𝐴𝐸 ||𝐵𝐶
̅̅̅̅ 𝑎𝑛𝑑 ̅̅̅̅
𝐴𝐶
𝐵𝐸 𝑖𝑛𝑡𝑒𝑟𝑠𝑒𝑐𝑡 𝑎𝑡 𝐷
2) < 𝐵 ≅ < 𝐸
2) Parallel Lines cut by a transversal form congruent
alternate interior angles
< 𝐴 ≅< 𝐶
3) ΔADE ~ ΔCDB
3) 𝐴𝐴~𝐴𝐴
 In this proof, you could have named only ONE set of alternate interior angles, and then used vertical
angles for the second angle.
#2)
Statement
Reason
1) < 𝐴 ≅ < 𝐶
̅̅̅̅
̅̅̅̅ ⊥ 𝐵𝐶
𝐷𝐸
̅̅̅̅
̅̅̅̅
𝐷𝐹 ⊥ 𝐴𝐵
2) < 𝐷𝐸𝐶 = 90°
< 𝐷𝐹𝐴 = 90°
3) < 𝐷𝐸𝐶 ≅ < 𝐷𝐹𝐴
4) 𝛥𝐴𝐹𝐷 ~ 𝛥𝐶𝐸𝐷
1) Given
2) Definition of perpendicular lines
3) All right angles are congruent
4) 𝐴𝐴~𝐴𝐴
#3)
Statement
1) < 𝐵𝐶𝐹 ≅ < 𝐶𝐸𝐷
2) < 𝐵𝐹𝐶 ≅ < 𝐷𝐹𝐸
3) ΔBCF ~ ΔDEF
Reason
1) Given
2) Vertical angles
3) 𝐴𝐴~AA
#4)
Statement
̅̅̅̅
̅̅̅̅
1) 𝐵𝐸 ⊥ 𝐴𝐶
̅̅̅̅ ⊥ 𝐴𝐵
̅̅̅̅
𝐶𝐷
2) < 𝐻𝐸𝐶 = 90°
< 𝐻𝐷𝐵 = 90°
3) < 𝐻𝐸𝐶 ≅ < 𝐻𝐷𝐵
4) < 𝐷𝐻𝐵 ≅ < 𝐸𝐻𝐶
5) ΔEHC ~ ΔDHB
Reason
1) Given
2) Definition of perpendicular lines
3) All right angles are congruent
4) Vertical angles
5) 𝐴𝐴~𝐴𝐴
#5)
Statement
̅̅̅̅ bisects < 𝐻𝐴𝐸
1) 𝐴𝐶
̅̅̅̅
𝐴𝐸 ≅ ̅̅̅̅
𝐴𝐻
2) < 𝐻𝐴𝐺 ≅ < 𝐸𝐴𝐺
3) ̅̅̅̅
𝐴𝐺 ≅ ̅̅̅̅
𝐴𝐺
4) ΔHAG ~ ΔEAG
Reason
1) Given
2) Definition of Angle Bisector
3) Reflexive Property
4) 𝑆𝐴𝑆 ~𝑆𝐴𝑆
6) (4)
7) (2)
8) a) <COD
c) <BOD
b) <AOE or <DOC
d) <AOC or <DOE
Similar Triangle Proofs (2) ANSWER KEY
#1) First identify what we are aiming for: ∆𝑪𝑭𝑫~∆𝑪𝑬𝑩
Statement
̅̅̅̅
1) < 𝐵 ≅< 𝐷, 𝐶𝐸 ⊥ ̅̅̅̅
𝐴𝐵, ̅̅̅̅
𝐶𝐹 ⊥ ̅̅̅̅
𝐴𝐷
2) < 𝐶𝐸𝐵 = 90°
< 𝐶𝐹𝐷 = 90°
3) < 𝐶𝐸𝐵 ≅ < 𝐶𝐹𝐷
4) ∆𝐶𝐹𝐷 ~ ∆𝐶𝐸𝐵
5)
𝐶𝐹
𝐶𝐸
=
𝐹𝐷
𝐸𝐵
Reason
1) Given
2) Definition of perpendicular lines
3) All right angles are congruent
4) 𝐴𝐴~𝐴𝐴
5) Corresponding sides of similar triangles are in proportion
#2) First identify what we are aiming for: ∆𝑭𝑩𝑪~∆𝑭𝑫𝑬
Statement
1) < 𝐸𝐵𝐶 ≅< 𝐶𝐷𝐸
2) < 𝐵𝐹𝐶 ≅ < 𝐷𝐹𝐸
3) ∆𝐹𝐵𝐶 ~ ∆𝐹𝐷𝐸
4)
𝐹𝐵
𝐹𝐷
=
Reason
1) Given
2) Vertical angles
3) 𝐴𝐴~𝐴𝐴
4) Corresponding sides of similar triangles are in proportion
𝐹𝐶
𝐹𝐸
#3) First identify what we are aiming for: ∆𝒀𝑻𝑿~∆𝒀𝑹𝒁
Statement
1) Isosceles ∆𝑊𝑋𝑍 with vertex W
< 𝑌𝑇𝑊 ≅< 𝑌𝑅𝑊
2) < 𝑊𝑋𝑍 ≅ < 𝑊𝑍𝑋
3) < 𝑌𝑇𝑊 & < 𝑌𝑇𝑋 are supplementary
< 𝑌𝑅𝑊 & 𝑌𝑅𝑍 are supplementary
4) < 𝑌𝑇𝑋 ≅< 𝑌𝑅𝑍
5) ∆𝑌𝑇𝑋 ~ ∆𝑌𝑅𝑍
𝑌𝑇
𝑋𝑌
6) 𝑌𝑅 = 𝑍𝑌
1) Given
2) Isosceles triangle angles
3) Supplementary Angles form linear pairs
4) Suppl of ≅ ∠𝑠 are ≅
5) 𝐴𝐴~𝐴𝐴
6) Corresponding sides of similar triangles are in proportion
2
5) 𝑦 − 5 = − 3 (𝑥 − 6)
4) (3)
6) x = 28
Reason
<C = 48
2
or 𝑦 = − 3 𝑥 + 9
Similar Triangle Proofs (3) ANSWER KEY
#1) First identify what we are aiming for: ∆𝑸𝑺𝑹~∆𝑷𝑴𝑹
Statement
̅̅̅̅
̅̅̅̅̅
1) 𝑄𝑆 and 𝑃𝑀 are altitudes
2) <𝑄𝑆𝑅 is a right angle
< 𝑃𝑀𝑅 is a right angle
3) < 𝑄𝑆𝑅 ≅ < 𝑃𝑀𝑅
4) < 𝑅 ≅< 𝑅
5) ∆𝑄𝑆𝑅 ~ ∆𝑃𝑀𝑅
𝑄𝑆
6) 𝑃𝑀 =
Reason
1) Given
2) Definition of Altitude
3) All right angles are congruent
4) Reflexive Property
5) 𝐴𝐴~𝐴𝐴
6) Corresponding sides of similar triangles are in proportion
𝑄𝑅
𝑃𝑅
7) 𝑄𝑆 ∙ 𝑃𝑅 = 𝑃𝑀 ∙ 𝑄𝑅
7) In a proportion, the product of the means equals the product
of the extremes
#2) First identify what we are aiming for: ∆𝑨𝑫𝑬~∆𝑨𝑪𝑩
Statement
̅̅̅̅ // 𝐸𝐷
̅̅̅̅
1) 𝐵𝐶
2) < 𝐴𝐷𝐸 ≅ < 𝐴𝐶𝐵
3) < 𝐴 ≅ < 𝐴
4) ∆𝐴𝐷𝐸 ~ ∆𝐴𝐶𝐵
𝐴𝐷
5) 𝐴𝐵 =
𝐷𝐸
𝐵𝐶
Reason
1) Given
2) Parallel lines cut by a transversal form congruent corresponding angles
3) Reflexive Property
4) 𝐴𝐴~𝐴𝐴
5) Corresponding sides of similar triangles are in proportion
#3) First identify what we are aiming for: ∆𝑨𝑩𝑪~∆𝑬𝑫𝑪
Statement
̅̅̅̅̅// 𝐷𝐸
̅̅̅̅
1) 𝐴𝐵
2) < 𝐵𝐴𝐶 ≅ < 𝐷𝐸𝐶
< 𝐴𝐵𝐶 ≅ < 𝐸𝐷𝐶
4) ∆𝐴𝐵𝐶 ~ ∆𝐸𝐷𝐶
5)
𝐴𝐵
𝐸𝐷
=
𝐴𝐶
𝐸𝐶
Reason
1) Given
2)Parallel Lines cut by a transversal form congruent alternate interior angles
4) 𝐴𝐴~𝐴𝐴
5) Corresponding sides of similar triangles are in proportion
6) 𝐴𝐵 ∙ 𝐸𝐶 = 𝐸𝐷 ∙ 𝐴𝐶
6) In a proportion, the product of the means equals the product of the extremes
 In this proof, you could have named only ONE set of alternate interior angles, and then used vertical
angles for the second angle.
#4)
Statement
̅̅̅̅
1) 𝐵𝐷 ⊥ ̅̅̅̅
𝐴𝐶
< 𝐵𝐴𝐷 ≅ < 𝐶𝐵𝐷
2) < 𝐵𝐷𝐴 = 90°
< 𝐶𝐷𝐵 = 90°
3) < 𝐵𝐷𝐴 ≅ < 𝐶𝐷𝐵
4) ∆𝐴𝐵𝐷 ~ ∆𝐵𝐶𝐷
Reason
1) Given
2) Definition of perpendicular lines
3) All right angles are congruent
4) 𝐴𝐴~𝐴𝐴
6) Area of ∆𝑋𝑌𝑍 = 12
5) x = 20
1
2
7) 𝑦 + 1 = (𝑥 − 4)
1
2
or 𝑦 = 𝑥 − 3