How do the corollaries to the Isosceles Triangle Theorem and it's converse follow from the theorems?

Question

I am working with Isosceles Triangles and corollaries. This is just one question I really don't understand. Thanks for your help!

Answer 1

The theorems and corollaries I can think of are:

1. If two sides of a triangle are of equal length then the two angles they make with a common base are also equal. The triangle is isosceles.
2. If two angles of a triangle are equal then the sides forming these angles have equal length. The triangle is isosceles.

The converses and their corollaries are: if no sides of a triangle are equal in length then none of its angles are equal and it is not an isosceles triangle. Also, if no angles of a triangle are equal, none of its sides have equal length and it is not an isosceles triangle. The triangle is scalene.

An isosceles triangle with all three angles the same is an equilateral triangle. An isosceles triangle with all sides the same length is an equilateral triangle.

The corollaries usually follow through proofs using congruency of triangles, more specifically by splitting the isosceles triangle into two back-to-back right-angled triangles, which are congruent.