Congruency: Congruent Triangles

130. AAA: The Size Trap · Same angles, different sizes

Equal angles guarantee similarity (same shape, sides in a fixed ratio) — never congruence. △DEF stretches and shrinks freely while its angles stay locked, so AAA cannot fix the size.

AAA is NOT a congruence criterion. Definition. If the three angles of one triangle are equal to the three angles of another triangle, the two triangles are *similar* — written △ABC ~ △DEF — meaning they have the same shape but not necessarily the same size. Their corresponding sides are in a fixed ratio (k = DE/AB = EF/BC = DF/AC), called the scale factor, which can be any positive number. Why AAA fails for congruence. Congruence requires both shape AND size to match. Three angles fix the shape (the triangle's *form*) because the angles sum to 180° — once two angles are given the third is determined, and the angles alone decide the triangle's proportions. But the angles place no constraint on the absolute side lengths. You can scale a triangle to any size and the angles never change. So infinitely many non-congruent triangles share the same three angles. Rule. To upgrade similarity to congruence, at least one pair of corresponding sides must be equal — this is what the SSS, SAS, ASA, AAS and RHS tests each provide. > Similar: equal angles, sides in the same ratio. > Congruent: equal angles and equal corresponding sides.

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Selina ICSE: Congruency: Congruent Triangles

What this lesson covers

Try to break it

Drag E to scale △DEF up or down. Whatever the size, ∠D = ∠A = 65°, ∠E = ∠B = 65°, ∠F = ∠C = 50° — the angles never change. But DE, EF, FD can be 1×, 2×, or 5× the matching sides of ABC. Same angles, scalable sides — that's why AAA is a similarity test, not a congruency test.

How you build it

Construct two triangles with the SAME three angles (65°, 65°, 50°) but DIFFERENT base lengths.

Draw a SHORT horizontal base AB (around 150 long). This is the first triangle.
At A, dial the angle to 65°. With the Angle tool, click A first, then click B.
At B, dial the angle to 65° (the SAME as ∠A — that makes △ABC isoceles). Click B first, then A.
Mark point C where the two rays cross. △ABC is complete: ∠A = ∠B = 65°, ∠C = 50°.
Now draw a LONGER base DE (around 300, twice as long as AB). This is the second triangle, with the SAME angles as ABC but a bigger size.
At D, dial to 65° — the SAME as ∠A. Click D first, then E.
At E, dial to 65° — the SAME as ∠B. Click E first, then D.
Mark point F where the two rays cross. △DEF is complete — same three angles as △ABC (65°, 65°, 50°), but the sides are roughly twice as long. AAA matched the SHAPE; the SIZE is different. That's why AAA is a similarity test, not a congruency test.

The proof, step by step

Prove that equal angles alone do not make two triangles congruent.

AAA ensures that the shape of the triangle is fixed, but not its size.
Congruency requires both shape AND size to be identical.
Since we can scale a triangle without changing its angles, AAA cannot prove congruency.

Worked example

Two triangles have all three corresponding angles equal. Are they necessarily congruent?

AAA guarantees similarity (same shape), but not congruency (same size). The triangles can be different sizes.

Yes, always
No, they are only similar — correct
Yes, if they share a side
No, they must have different angles