If two sides of a triangle are <span>congruent </span>, then the angles opposite to these sides are congruent.
<span><span>∠P≅∠Q</span><span>∠P≅∠Q</span></span>
Proof:
Let <span>SS</span> be the midpoint of <span><span><span>PQ</span><span>¯¯¯¯¯</span></span><span><span>PQ</span>¯</span></span> .
Join <span>RR</span> and <span>SS</span> .
Since <span>SS</span> is the midpoint of <span><span><span>PQ</span><span>¯¯¯¯¯</span></span><span><span>PQ</span>¯</span></span> , <span><span><span><span>PS</span><span>¯¯¯¯¯</span></span>≅<span><span>QS</span><span>¯¯¯¯¯</span></span></span><span><span><span>PS</span>¯</span>≅<span><span>QS</span>¯</span></span></span> .
By <span>Reflexive Property </span>,
<span><span><span><span>RS</span><span>¯¯¯¯¯</span></span>≅<span><span>RS</span><span>¯¯¯¯¯</span></span></span><span><span><span>RS</span>¯</span>≅<span><span>RS</span>¯</span></span></span>
It is given that <span><span><span><span>PR</span><span>¯¯¯¯¯</span></span>≅<span><span>RQ</span><span>¯¯¯¯¯</span></span></span><span><span><span>PR</span>¯</span>≅<span><span>RQ</span>¯</span></span></span>
Therefore, by <span>SSS </span>,
<span><span>ΔPRS≅ΔQRS</span><span>ΔPRS≅ΔQRS</span></span>
Since corresponding parts of congruent triangles are congruent,
<span><span>∠P≅∠Q</span><span>∠P≅∠Q</span></span>
The converse of the Isosceles Triangle Theorem is also true.
If two angles of a triangle are congruent, then the sides opposite those angles are congruent.
If <span><span>∠A≅∠B</span><span>∠A≅∠B</span></span> , then <span><span><span><span>AC</span><span>¯¯¯¯¯</span></span>≅<span><span>BC</span><span>¯¯¯¯¯</span></span></span><span><span><span>AC</span>¯</span>≅<span><span>BC</span>¯</span></span></span> .
