## Special Features of Isosceles Triangles

Isosceles triangles are special and because of that there are unique relationships that involve their internal line segments. Consider isosceles triangle *ABC *in Figure 1.

* *

**Figure 1 ***An isosceles triangle with a median.*

With a median drawn from the vertex to the base, *BC* , it can be proven that Δ *BAX* ≅ Δ *CAX*, which leads to several important theorems.

*Theorem 32:* If two sides of a triangle are equal, then the angles opposite those sides are also equal.

*Theorem 33:* If *a triangle is equilateral, then it is also equiangular. *

*Theorem 34:* If two angles of *a triangle are equal, then the sides opposite these angles are also equal. *

*Theorem 35:* If a triangle is equiangular, then it is also equilateral.

**Example 1:** Figure has Δ *QRS* with *QR* = *QS.* If *m* ∠ *Q* = 50°, find *m* ∠ *R* and *m* ∠ *S.*

* *

**Figure 2 **An isosceles triangle with a specified vertex angle.** **

Because *m* ∠ *Q* + *m* ∠ *R* + *m* ∠ *S* = 180°, and because *QR* = *QS* implies that *m* ∠ *R* = *m* ∠ *S*,

**Example 2:** Figure 3 has Δ *ABC* with *m* ∠ *A* = *m* ∠ *B* = *m* ∠ *C*, and *AB* = 6. Find *BC* and *AC.*

* *

**Figure 3 ***An equiangular triangle with a specified side.*

*
*

Because the triangle is equiangular, it is also equilateral. Therefore, *BC* = *AC* = 6.