127. ASA Congruence Criterion · Two angles and the included side lock the triangle

Try to break it

Drag B to change side BC's length. As long as the two given angles and the included side BC stay matched in both triangles, the triangles stay congruent. ASA refuses to break.

How you build it

Construct a triangle given ASA data.

• Draw side AB of length c — the given included side. Segment tool: click for A, then click for B (to the right of A, so AB runs left-to-right).
• At A, set the dial to the first given angle. With the Angle tool, click A first, then click B — this draws a ray from A measured anticlockwise from AB. The ray opens above AB.
• At B, set the dial to the second given angle. With the Angle tool, click B first, then click A — this draws a ray from B measured from BA, opening above AB on the same side as the first ray. The two rays meet at C, completing △ABC.
• Mark point C where the two rays cross. The Point tool will snap to the intersection. △ABC is now constructed by ASA — and any triangle built from the same side AB and the same two angles at A and B is congruent to this one.

The proof, step by step

Prove that two angles and the included side make the triangles congruent (ASA).

• In ΔABC and ΔPQR, AB = PQ (Given)
• ∠A = ∠P (Given)
• ∠B = ∠Q (Given)
• ∴ ΔABC ≅ ΔPQR (by ASA Congruence Rule)

Worked example

In ΔABC and ΔDEF, AB = DE = 6 cm, ∠A = ∠D = 50°, and ∠B = ∠E = 80°. Which congruence condition proves that ΔABC ≅ ΔDEF?

Two angles and the included side are equal, so the ASA congruence condition applies.

• SAS (Side-Angle-Side)
• ASA (Angle-Side-Angle) — correct
• SSS (Side-Side-Side)
• RHS (Right angle-Hypotenuse-Side)