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?
