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Geometry & measures · Congruence & similarity

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Congruent triangles & proof

The four congruence conditions SSS, SAS, ASA/AAS and RHS, why SSA is not one of them, and how to write a congruence proof that earns every mark.

Geometry & measures · Congruence & similarity

Congruent triangles & proof

The four congruence conditions SSS, SAS, ASA/AAS and RHS, why SSA is not one of them, and how to write a congruence proof that earns every mark.

Why it works

Congruent means identical — same shape, same size. One shape can be picked up, turned, flipped and laid exactly on the other. Reflections, rotations and translations all produce congruent images; enlargements (except by factor ±1\pm 1) do not.

You never need all six measurements. A triangle has three sides and three angles, but the right three facts pin it down completely. There are exactly four such sets:
ConditionWhat you need
SSSall three pairs of sides equal
SAStwo sides and the angle between them
ASA / AAStwo angles and a corresponding side
RHSright angle, hypotenuse and one other side
Why the A in SAS must be between the two sides. Fix two side lengths and the angle between them and there is only one way to close the triangle. But give the angle away from one of the sides — "SSA" — and there can be two different triangles that fit: the third side can swing to meet the base in two places. That's why SSA is not a congruence condition, and why "AAA" fails too: equal angles force the same shape but any size, which is similarity, not congruence.

Writing the proof: three sides of the argument. A congruence proof earns marks for structure, not just for the right answer:
  1. State each equal pair with its reason — "AB=DEAB = DE (given)",
"angle BAC=BAC = angle EDFEDF (alternate angles are equal)", "BC=EFBC = EF (sides of a square)".
  1. Name the condition — "so triangles ABCABC and DEFDEF are congruent
(SAS)".
  1. Get the letters in corresponding order. Writing
△ABC≡△DEF\triangle ABC \equiv \triangle DEF claims AA matches DD, BB matches EE and CC matches FF. Scrambled letters lose the mark even when the triangles really are congruent.

Then use it. Once triangles are proved congruent, every pair of corresponding sides and angles is equal — which is usually how the question finishes ("hence show that PQ=RSPQ = RS").