Similar or Congruence Triangles Theorem Proof

The theorem states that the two triangles are said to be similar if the corresponding sides and their angles are equal or congruent.

Similar or Congruence Triangles Theorem

If the corresponding sides of two triangles are proportional, then their corresponding angles are equal and the triangles are similar.
Two triangles ABC and DEF Such that AB/DE = BC/EF = AC/DF

To prove:
ABC similar to DEF

Mark points P and Q on DE and DF respectively such that DP = AB and DQ = AC, join PQ.

Since AB/DE = AC/DF and AB = DP, AC = DQ Therefore DP/DE = DQ/DF.
This implies that PQ is parallel to EF
So, < DPQ =< E and < DQP =< F (corresponding angles are equal)
Triangle DPQ similar to triangle DEF (By AA similarity) --(1)

So, DP/DE = PQ/EF Or AB/DE = PQ/EF (since DP = AB)
This implies that BC = PQ

Now AB = DP, AC = DQ and BC = PQ
This implies that ABC and DPQ (SSS congruence) --2)

From 1 and 2, we get
ABC similar to DEF
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The online proof of finding the similar or congruence triangles is made easier here.

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