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If the base and a leg of one isosceles triangle are congruent to the base and a leg
of another isosceles triangle, then their vertex angles are congruent.
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In any isosceles triangle, the legs are congruent.
So, if one leg of one isosceles triangle is congruent to one leg
of the other isosceles triangle, then the other leg of the first
isosceles triangle is congruent to the other leg of the second isosceles triangle.
Thus the sides of one triangle are congruent to the sides of the other triangle.
According to the SSS test of congruency for triangles, the triangles themselves are congruent.
Hence, their vertex angles are congruent, since these angles are corresponding angles.
At this point, the proof is complete.
Solved.