Download Proving Triangles are Congruent

Proving Triangles are Congruent
(NOTES)
There should be 5 statements with
Given: 𝐴𝐶 ≅ 𝐷𝐵, ∠𝐴𝐶𝐷 ≅ ∠𝐵𝐷𝐶
justification
Prove: ∠𝐴 ≅ ∠𝐵
• Statement #1 is given.
• Statement # 2 is also given.
• Statement # 3
𝐴
𝐶
– Reflexive Property
• Triangles Sharing A Side
– Vertical Angles
• Triangles With angles facing each
other
– Mid-Point
•
• Mid-point is given, triangles
connected by a point.
Statement # 4 Congruence Postulates
𝐷
𝐵
Statements
Justifications
1. 𝐴𝐶 ≅ 𝐷𝐵
Given
2.∠𝐴𝐶𝐷 ≅ ∠𝐵𝐷𝐶
Given
3. 𝐶𝐷 ≅ 𝐶𝐷
Reflexive Property
4. ∆𝐴𝐷𝐶 ≅ ∆𝐵𝐶𝐷
SAS Congruency Postulate
– SSS, SAS, AAS, ASA
• Statement # 5 (CPCTC)
– Corresponding Parts of Congruent
Triangles are Congruent
5.
∠𝐴 ≅ ∠𝐵
CPCTC
Prove: M  I
 IA
Given: MN  IN
Given: MA
M
N
A
I
Statement
 IA
 IN
3.) NA  NA
4.) MAN  IAN
1.) MA
2.) MN
5.) M  I
Justification
Given
Given
Reflexive Property
SSS
CPCTC
Prove:
N
𝐴𝐷 ≅ 𝐸𝐶
Given: B is the midpoint of 𝐷𝐶
A
Statement
Justification
 𝐸𝐵 Given
2.) 𝐷𝐵  𝐶𝐵 Given (Definition of Midpoint)
3.) ∠𝐷𝐵𝐴  ∠𝐶𝐵𝐸 Vertical Angles
4.) ∆𝐷𝐵𝐴  ∆𝐶𝐵𝐸 SAS
1.)
𝐴𝐵
5.)
𝐴𝐷 ≅ 𝐸𝐶
CPCTC
Prove:
T
∠𝑇 ≅ ∠𝑃
Given: R is the midpoint of 𝑆𝐼
R
I
S
Statement
P
 ∠I
2.) 𝑆𝑅  𝐼𝑅
3.) ∠𝑇𝑅𝑆  ∠𝑃𝑅𝐼
4.) ∆𝑇𝑅𝑆  ∆𝑃𝑅𝐼
1.)
5.)
∠𝑆
∠𝑇 ≅ ∠𝑃
Justification
Given
Given (Definition of Midpoint)
Vertical Angles
ASA
CPCTC
Prove: ∠𝑂 ≅ ∠P
Statement
Justification
Prove:
∠𝑄 ≅ ∠𝑁
Prove:
∠𝐴 ≅ ∠𝐶
Prove: ∠𝑅 ≅ ∠𝑃
R
Statement
T
I
P
SHOW ALL YOUR WORK
Justification
Prove: ∠𝐾 ≅ ∠M
Statement
SHOW ALL YOUR WORK
Justification
Prove:
∠𝐷 ≅ ∠𝐸
Prove:
∠𝐺 ≅ ∠I
Prove: ∠𝑂 ≅ ∠𝑃
Prove: ∠𝑆 ≅ ∠𝑊
Given: U is the midpoint of 𝑆𝑊
Prove:
∠𝐿 ≅ ∠𝐽
Prove:
∠𝑋 ≅ ∠𝑅
T
I
R
Prove: ∠𝑇 ≅ ∠𝑅
T
P
Statement
I
R
SHOW ALL YOUR WORK
Justification
In ∆MON the 𝑚∠𝑀 53° & 𝑚∠O 90°. Write the sides of the ∆ in descending order:
In ∆ROD the 𝑚∠𝑅 46° & 𝑚∠O 95°. Write the sides of the ∆ in descending order:
If two sides of a triangle re 9 and 17, which of these can not be the third side
of the triangles? Explain why? 9, 18, 21, 26, 35
If two sides of a triangle re 21 and 30, which of these can not be the third side
of the triangles? Explain why? 51, 10, 18, 49, 53, 12
In triangle PQR, PQ>QR & QR > RP. Which angle in triangle PQR has the
smallest measure
In ∆MON the 𝑚∠𝑀 60° & 𝑚∠O 82°. Write the sides of the ∆ in descending order:
In ∆ROD the 𝑚∠𝑅 63° & 𝑚∠O 105°. Write the sides of the ∆ in descending order:
If two sides of a triangle are 32 and 9, which of these can not be the third side
of the triangles? Explain why? 9, 25, 28, 26, 41
If two sides of a triangle re 18 and 35, which of these can not be the third side
of the triangles? Explain why? 53, 50, 18, 20, 17, 99
In triangle RST, RS>ST & ST > TR. Which angle in triangle PQR has the smallest
measure