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 Description  
 Side-Angle-Side (SAS):  A congruence property that states if two sides and the included angle of a triangle are congruent, respectively, to two sides and the included angle of another triangle, then the two triangles are congruent.

The above picture shows two congruent triangles according to the SAS congruence property. In the picture, the triangle ABC is isosceles (Sides AB and AC are congruent). If we construct the angle bisector for A (indicated by P), we get two congruent angles, angle CAP and angle BAP. The segment AP is congruent with itself. Thus, we have two sides and the included angle in one triangle being congruent with the corresponding parts of the other triangle. So, by SAS triangles CAP and BAP are conrguent.

In his books Elements, Euclid showed that for SAS, ASA, AAS and SSS, if three corresponding parts of two triangles are congruent, then all six parts have corresponding congruent parts. However, there are instances in which five of the six parts of one triangle are congruent to five of the six parts of another triangle without the triangles being congruent! Consider the following pair of triangles. Five parts of one triangle are congruent to five parts of the second triangle. But why are they not congruent?