1. 4.3 – Triangle Congruence by ASA and AAS
By Andy Neale

2. What are ASA and AAS?
ASA and AAS are both ways to prove that two triangles are equal.
ASA stands for Angle-Side-Angle.
AAS stands for Angle-Angle-Side.

3. ASA Postulate
This means that if, in a triangle, the measures of an angle, a segment that is adjacent to that angle, and the other angle adjacent to that segment are congruent to those corresponding angles and sides of another triangle, then the two triangles are congruent. This always has to be in the order of angle, then side, then angle.

4. ASA in a picture
Triangle ABC is congruent to triangle A’B’C’ by ASA. Angle A is congruent to angle A’, segment AC is congruent to segment A’C’, and angle C is congruent to angle C’.

5. Can you use ASA?
Using what we have learned already about congruent angles, there is a way to prove that triangle NML is congruent to triangle NPO.

6. Solution to the example
Angle MNL is congruent to angle PNO because vertical angles are congruent.
Segment NM is congruent to segment NP (this is given).
Angle M is congruent to angle P (this is also given).
Therefore, triangle NML is congruent to triangle NPO by the ASA Postulate.

7. AAS Theorem
This means that if, in a triangle, the measures of an angle, another angle, and the segment adjacent to that angle (but not between the two angles) are congruent to those corresponding angles and sides of another triangle, then the two triangles are congruent. This always has to be in the order of angle, then angle, then side.

8. AAS in a picture
Triangle ABC is congruent to triangle A’B’C’ by AAS. Angle B is congruent to angle B’, angle A is congruent to angle A’, and segment AC is congruent to segment A’C’.

9. Can you use AAS?
Using what we have learned already about congruent angles, there is a way to prove that triangle SRP is congruent to triangle QRP.

10. Solution to the example
Angle S is congruent to angle Q (this is given).
Angle SRP is congruent to angle QRP, because of the definition of an angle bisector.
Segment RP is congruent to segment RP by the reflexive property of congruence.
Therefore, triangle SRP is congruent to triangle QRP by AAS.

11. Congratulations!
You should now understand the ASA postulate and the AAS theorem, what they mean, and how they work.
Hopefully this helps for anybody who needs it to study.