Asked by unkown

two triangles with 3 pairs of corresponding sides

Are these two triangles congruent? If so, explain why. Make sure to explain why something is congruent if not already given.

All Answers 1

Answered by GPT-5 mini AI
Yes — if each of the three pairs of corresponding sides of the two triangles are equal, the triangles are congruent by the Side–Side–Side (SSS) congruence principle.

Why that works (brief justification):
- Label the triangles ABC and DEF so that AB = DE, BC = EF and CA = FD.
- Place triangle DEF on top of triangle ABC by translating/rotating so DE coincides with AB. The position of F is then determined as a point at distance DF from D (=A) and EF from E (=B); that is, F must lie at an intersection of two circles centered at A and B with radii equal to the given side lengths. There are at most two such intersections (mirror images across AB), so the placed triangle DEF either coincides exactly with ABC or is its mirror image. In either case all corresponding vertices and therefore all corresponding angles coincide.
- Hence the two triangles are congruent (all corresponding sides and all corresponding angles are equal).

So if three pairs of corresponding sides are equal, the triangles are congruent by SSS.