208. AAS Congruence Criterion · Two Angles, One Side

Try to break it

Drag A or C into any triangle you like. As long as DEF shares ABC's two angles and side BC, it stays perfectly congruent — same AB, same AC, every time. Two angles and a non-included side fix a triangle completely, so the match cannot be broken.

How you build it

Build two triangles from two angles and a non-included side (AAS).

• Point tool: mark point B on the right half of the canvas.
• Point tool: mark point C to the right of B — BC is the given non-included side.
• Segment tool: join B to C.
• Angle tool: set the dial to 70°, click B, then click toward C. This arm becomes side BA.
• Angle tool: set the dial to 60° (= 180 − 50 − 70), click C, then click toward B. This arm becomes side CA; the two arms cross at A.
• Point tool: mark point A where the two arms cross — ΔABC is complete.
• Now the second triangle. Point tool: mark point E on the left half of the canvas.
• Ray tool: click E, then click further right — EF will lie on this ray.
• Arc tool: click centre B, drag until the arc snaps to C. This locks the compass to the exact length BC.
• Arc tool: click centre E. With length BC locked, this arc crosses the base ray — that crossing is F, so EF = BC.
• Point tool: mark F where the arc crosses the base ray.
• Angle tool: set the dial to 70° (= ∠B), click E, then click toward F. This arm becomes side ED.
• Angle tool: set the dial to 60° (= ∠C), click F, then click toward E. This arm becomes side FD; the two arms cross at D.
• Point tool: mark point D where the two arms cross. Same two angles, same side (BC = EF) — so ΔDEF ≅ ΔABC. That is AAS!

The proof, step by step

Prove that two angles and a non-included side make the triangles congruent (AAS).

• In Δ ABC and Δ DEF, ∠A = ∠D and ∠B = ∠E (Given)
• We know that ∠A + ∠B + ∠C = 180° and ∠D + ∠E + ∠F = 180° (Angle Sum Property)
• Therefore, ∠C = ∠F (Since ∠A=∠D and ∠B=∠E)
• In Δ ABC and Δ DEF: ∠B = ∠E, BC = EF (Given), ∠C = ∠F (Proved above)
• ∴ Δ ABC ≅ Δ DEF (By ASA Congruency Rule)

Worked example

In Δ PQR and Δ XYZ, ∠P = ∠X = 50°, ∠Q = ∠Y = 70°, and QR = YZ = 8 cm. Which congruence criterion proves Δ PQR ≅ Δ XYZ?

We are given two angles (∠P=∠X, ∠Q=∠Y) and a non-included side (QR=YZ). Since two angles are equal, the third angle is also equal, making it equivalent to ASA. The direct criterion for two angles and a non-included side is AAS.

• SSS (Side-Side-Side)
• SAS (Side-Angle-Side)
• ASA (Angle-Side-Angle)
• AAS (Angle-Angle-Side) — correct