If you know that two angles and the included side are congruent to the corresponding side and angles of another triangle, then you can use the ASA congruence postulate directly.
If you know two corresponding angles are congruent in a pair of triangles AND one corresponding pair of sides NOT between those two angles are congruent, then you can apply the AAS congruence postulate.
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AB A' B'
Suppose ∠A ≡ ∠A' and ∠B ≡ ∠B", and the line segment at the bottoms of the above triangles are congruent, then you can use ASA directly, as indicated above.
But if any other pair of sides are congruent, you can apply AAS. AAS works because if two pairs of angles on two different triangles are congruent, then the other pair of angles is also congruent. The sum of the interior angles in any triangle measures a total of 180 degrees. So the other angle must measure 180 degrees - m∠A -m∠B. You now have ASA, since whereever the side is, it now must be in between a pair of congruent angles.
Because we can show that AAS always works, we don't need to go through the details each time of showing why it works. We just need to show that one pair of angles in each of two triangles is congruent, and that one corresponding side in each triangle is congruent but is not between the angles known to be congruent.
James S.
10/25/23