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.
Special Features of Isosceles Triangles
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.