# About congruence of two triangles

In Euclidean geometry, we know that SSS, SAS, AAS (or equivalently ASA) and RHS are the only 4 tools for proving the congruence of two triangles. I am wondering if the following can be added to that list:-

If ⊿ABC~⊿PQR and the area of ⊿ABC = that of ⊿PQR, then the two triangles are congruent.

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## 2 Answers

If two triangles (in fact, any two polygons) are similar, with scale factor $k$, then the ratio of their areas is $k^2$. If the areas are equal then $k^2=1$ which implies that $k=1$, and therefore the triangles are congruent.

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Yes, the statement holds true; equality of the three angles and the area of the two triangles implies congruence. The area of a triangle is dependent of the three sides, so one can use Heron's formula to proof this statement.

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