Special Side Arrangement

Geometry Level 3

△ A B C \triangle ABC and △ D E F \triangle DEF are such that A B = D E = 20 , AB = DE = 20, B C = E F = 18 , BC = EF = 18, and ∠ C = ∠ F . \angle C = \angle F. Can we conclude that △ A B C ≅ △ D E F \triangle ABC \cong \triangle DEF ?

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Steven Yuan by Steven Yuan , 20, USA

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1 solution

Jordan Cahn
Mar 1, 2018

In the given situation, we know two side congruencies and an angle congruency not contained by the two sides (SSA). Generally, this is insufficient to prove the triangles are congruent. This is because, when the non-adjacent side to the angle is shorter than the adjacent side, two triangles are possible:

However, when the non-adjacent side is longer than the adjacent side, only one triangle is possible:

Since this is the situation we are dealing with, the triangles must be congruent.

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