等腰三角形定理的逆向定理

Converse of isosceles triangle

As in properties of isosceles triangles (Isosceles Triangle Theorem), it's important to note that it is not immediately obvious that

1. A triangle has two congruent sides and
2. A triangle has two congruent angles

are equivalent statements.

In this section we do indeed want to show their equivalence.

So in particular, we would like to show

If a triangle has two congruent angles, then the sides opposite those angles are congruent.

Let's take a triangle $ABC$ with angles $B$ and $C$ equal.

As in the proof of the original Iscosceles Triangle Theorem, we want to draw the line to the midpoint of $BC.$ But we want to use the angle conditions so we will be careful in how we will define $D.$

Both of the above correct answers give us angle conditions involving $D$ which surely should help us. Now we'll just pick one way to define $D$, so let $D$ be the foot of the perpendicular from $A$ to $BC.$

Great! Here's what I came up with:

We know that $$\left.\begin{aligned} \angle ADC &= \angle ADB,\\ \angle ABD &= \angle ACD,\\ AD &= AD \\ \end{aligned} \right\} \Longrightarrow \triangle ADB\cong \triangle ADC$$ From which it follows that $AB = AC.$ So we showed that in a triangle with two congruent angles, the sides opposite those angles are congruent, as desired. :)