Triangles [Congruency in Triangles]

206. Same Shape, Same Size · The definition of congruent triangles

Corresponding sides and angles stay equal, so triangle DEF can always be slid to coincide exactly with ABC.

Two triangles are congruent (written ΔABC ≅ ΔDEF) when they have the same shape and size — every pair of corresponding sides is equal (AB = DE, BC = EF, CA = FD) and every pair of corresponding angles is equal. Congruent triangles can be superimposed, fitting exactly one over the other. Drag D to slide triangle DEF onto ABC — it covers it perfectly.

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For subscribers · Selina ICSE: Triangles [Congruency in Triangles]

What this lesson covers

Try to break it

Drag D to slide triangle DEF around. The three sides and three angles stay locked equal to those of ABC, so DEF can always land exactly on ABC. Try to find a position where DEF stops being congruent to ABC — impossible. Sliding preserves congruence.

How you build it

Construct two congruent triangles using the SSS rule.

Point tool: mark point A near the lower-left.
Point tool: mark point B to the right of A.
Point tool: mark point C above AB — this is your given triangle ABC (any shape works).
Segment tool: join A to B.
Segment tool: join B to C.
Segment tool: join C to A.
Point tool: mark point D well to the right — the start of your copy.
Ray tool: click D, then click further right — side DE will lie on this ray.
Arc tool: click centre A, drag until the arc snaps to B. This locks the radius to the exact length AB.
Arc tool: click centre D. With AB locked, the arc crosses the base ray — that crossing is E, so DE = AB.
Point tool: mark E where the arc meets the base ray.
Arc tool: click centre A, drag until the arc snaps to C. This locks the radius to AC.
Arc tool: click centre D — this arc (radius AC) sweeps above the base; F will lie on it.
Arc tool: click centre B, drag until the arc snaps to C. This locks the radius to BC.
Arc tool: click centre E — this arc (radius BC) crosses the D-arc at F.
Point tool: mark F where the two arcs meet, above the base.
Segment tool: join D to F.
Segment tool: join E to F. All three sides match (DE = AB, DF = AC, EF = BC) — so ΔDEF ≅ ΔABC by SSS.

The proof, step by step

Prove that the two triangles are congruent — all corresponding sides and angles equal.

DEF is a rigid copy of ABC, meaning it preserves all side lengths and angles.
Therefore, corresponding sides are equal: AB=DE, BC=EF, CA=FD.
If two triangles coincide when placed one over the other, they are congruent.

Worked example

Two triangles are congruent if they have the same shape and size. Which of the following must be true for congruent triangles?

Congruent triangles have exactly the same shape and size, meaning all corresponding sides and angles are equal.

Only their areas are equal
All corresponding sides and angles are equal — correct
Only their perimeters are equal
They must be right-angled triangles