Begin by drawing an isosceles triangle ABC:
Now I'll draw a green line AD, from the top vertex A to the bottom side BC, but
not perpendicular to BC, meeting BC at D, like this:
The two sides AB, AC, of the big isosceles triangle are congruent. The green
line AD is congruent to itself. The base angles B and C of the big isosceles
triangle ABC are congruent.
Therefore we have a case of SSA with triangles ABD and ACD. That is, two sides
and a nonincluded angle of one triangle, ABD, are congruent to the
corresponding two sides and angle of a second triangle ACD. However they are
not congruent since AD is not perpendicular to the base BC. Therefore the two
angles at D are supplementary but not congruent.
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However, the following version of the SSA theorem can be proved:
SSA Theorem: If two sides and a nonincluded angle of one triangle are
congruent to the corresponding two sides and angle of a second triangle, then
the triangles are either congruent or else the angles opposite the congruent
sides are supplementary.
Edwin
