Geometry: 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 2 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.

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