1. Triangle Congruence by ASA and AAS
Skill 21b

2. Objective
HSG-SRT.5: Students are responsible for proving two triangles are congruent by ASA and AAS.

3. Postulate 14 E B F C D A If ∠𝑨≌∠𝑫 , ∠𝑪≌∠𝑭 , and 𝑨𝑪 ≌ 𝑫𝑭 Then ∆𝑨𝑩𝑪≌∆𝑫𝑬𝑭
Angle-Side-Angle Postulate
If two angles and the included side of one triangle aer congruent to two angles and the included side of another triangle, then the two triangles are congruent. B A C E D F
If ∠𝑨≌∠𝑫
, ∠𝑪≌∠𝑭
, and 𝑨𝑪 ≌ 𝑫𝑭
Then ∆𝑨𝑩𝑪≌∆𝑫𝑬𝑭

4. Theorem 19 E B F C D A If ∠𝑨≅∠𝑫 , ∠𝑩≅∠𝑬 , and 𝑨𝑪 ≌ 𝑫𝑭 Then ∆𝑨𝑩𝑪≌∆𝑫𝑬𝑭
Angle-Angle-Side Theorem
If two angles and a nonincluded side of one triangle are congruent to two angles and the corresponding nonincluded side of another triangle, then the two triangles are congruent. B A C E D F
If ∠𝑨≅∠𝑫
, ∠𝑩≅∠𝑬
, and 𝑨𝑪 ≌ 𝑫𝑭
Then ∆𝑨𝑩𝑪≌∆𝑫𝑬𝑭

5. Theorem 19; AAS Theorem Given: ∠𝐴≅∠𝐷, ∠𝐵≅∠𝐸, and 𝐴𝐶 ≌ 𝐷𝐹
Prove: ∆𝐴𝐵𝐶≌∆𝐷𝐸𝐹 B A C E D F
Statement Reason
1) ∠𝐴≅∠𝐷, ∠𝐵≅∠𝐸,
& 𝐴𝐶 ≌ 𝐷𝐹
1) Given
2) ∠𝐶≅∠𝐹
2) 3rd Angle Thm.
3) ∆𝐴𝐵𝐶≌∆𝐷𝐸𝐹
3) ASA Postulate

6. Example 1; Writing a proof using ASA Post.
Given: 𝐴𝐵 ≌ 𝐷𝐸 , ∠𝐴≅∠𝐷, and ∠𝐵 & ∠𝐸 are rt. ∠’s.
Prove: ∆𝐴𝐵𝐶≌∆𝐷𝐸𝐹 A B C E D F
Statement Reason
1) 𝐴𝐵 ≌ 𝐷𝐸 , ∠𝐴≅∠𝐷, and ∠𝐵 and ∠𝐸 are right angles.
1) Given
2) ∠𝐵≌∠𝐸
2) ≌ of Rt. ∠’s Thm.
3) ∆𝐴𝐵𝐶≌∆𝐷𝐸𝐹
3) ASA Postulate

7. Example 2; Using SAS Postulate
a) Which triangles are congruent? Explain. S U V E N O T W A
∠𝐸≌∠𝑈
∠𝑂≌∠𝑉
𝐸𝑂 ≌ 𝑈𝑉 (included side)
∆𝐸𝑂𝑁≌∆𝑈𝑉𝑆 by SAS Postulate

8. Example 2; Using SAS Postulate
b) Which triangles are congruent? Explain. F N I O H G T C A
∠𝐶≌∠𝐺
∠𝐴≌∠𝐻
𝐶𝐴 ≌ 𝐺𝐻 (included side)
∆𝐶𝐴𝑇≌∆𝐺𝐻𝑂 by SAS Postulate

9. Example 3; Writing a proof using ASA Post.
Given: ∠𝐶𝐴𝐵≅∠𝐷𝐴𝐸, 𝐵𝐴 ≌ 𝐸𝐴 , and ∠𝐵 & ∠𝐸 are right angles.
Prove:
∆𝐴𝐵𝐶≌∆𝐷𝐸𝐹 A B C E D
Statement Reason
1) ∠𝐶𝐴𝐵≅∠𝐷𝐴𝐸, 𝐵𝐴 ≌ 𝐸𝐴 , and ∠𝐵 & ∠𝐸 are right angles.
1) Given
2) ∠𝐵≌∠𝐸
2) ≌ of Rt. ∠’s Thm.
3) ∆𝐴𝐵𝐶≌∆𝐷𝐸𝐹
3) ASA Postulate

10. #21b: Triangle Congruence by ASA & AAS
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