1. 4.5 Proving Triangles Congruent - ASA and AAS
*Then: You proved triangles congruent using SSS and SAS *Now: 1. Use the ASA Postulate to test for congruence. 2. Use the AAS Theorem to test for congruence.

2. Review SSS SAS Two ways to prove triangles are congruent-

3. Postulate 4.3: Angle-Side-Angle (ASA) Congruence
If two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the two triangles are congruent.
If Angle  A  D,
Side AC  DF, and
Angle  C  F,
then  ABC  DEF.

4. Example 1: Write a two-column proof
Given: AB  CD
ACD  CAB
Prove: ABC   CDA
Statements Reasons
1. AB  CD __________________
2. ACD  CAB 2. __________________
3. BCA  DAC 3. __________________
4. BD  BD __________________
5. ABC   CDA 5. __________________

5. Example 2: Write a flow proof
Given: S  V T is the midpoint of SV Prove:  RTS  UTV
S  V
T is the midpoint of SV
RTS  VTU
ST  TV
 RTS  UTV

6. Example 3: Write a paragraph proof
Given: CD bisects AE, AB  CD E  BCA Prove:  ABC  CDE It is _________ that ____ _____ and CD _____ AE. So, ____  _____ by definition of ____________. Also given that ___  ___, so ____ _____ by __________________________________________. Then  ____  _____ by _____________________

7. Theorem 4.5: Angle-Angle-Side (AAS) Congruence
If two angles and a nonincluded side of one triangle are congruent to two angles and the corresponding nonincluded side of a second triangle, then the two triangles are congruent.
If Angle  A  D,
Angle  C  F, and
Side BC  EF,
then  ABC  DEF.

8. Example 4: Complete the proof.
Given:  G  B, CB  GA
Prove: GCA  BAC
Statements Reasons
1. CB  GA 1. ___________________
2.  BCA  GAC 2. ___________________
3.  G  B 3. ___________________
4. AC  AC ___________________
5. GCA  BAC 5. ___________________

9. Example 5 : Write a flow proof
ADB  ACE
EC  DB
AEC  ABD
AEC  ABD
A  A

10. Example 6: Write a paragraph proof
Given:  S  U TR bisects STU Prove: SRT  URT It is given that ______bisects ______ so  _____  ______ by ______________________. Also given is ________________. _____  _____ by __________________________________________. Then _____  ______ by _______________________. Finally,  _____  ______ by ___________________ .

11. 4.5 Assignment:
p See handout