Isosceles Triangles Have Two Equal Sides

An isosceles triangle is defined by two sides of equal length. Those two equal sides are called the legs, and the third side is the base. The angle opposite the base — sitting between the two equal legs — is the vertex angle. The two angles at the base are the base angles, and here's the key property: base angles of an isosceles triangle are always equal.

That property works in reverse too. If two angles of a triangle are equal, the sides opposite them are equal — so the triangle must be isosceles.

Finding Angles

Example 1: The vertex angle of isosceles triangle ABC is 120°. Find each base angle.

Let \(x\) = the measure of each base angle. Since both base angles are equal:

$$x + x + 120 = 180$$ $$2x = 60$$ $$x = 30$$

Each base angle measures 30°.

Example 2: In isosceles triangle RST, the vertex angle S measures an unknown amount. The base angles R and T both measure 64°. Find ∠S.

$$64 + 64 + \angle S = 180$$ $$\angle S = 180 - 128 = 52°$$

Setting Up an Equation from a Word Problem

Some problems describe a relationship between the angles rather than giving you specific values. That just means you need to translate the words into an equation first.

Example 3: The measure of a base angle of isosceles triangle XYZ exceeds three times the vertex angle Y by 60°. Find the vertex angle.

Let \(Y\) = the vertex angle. Then each base angle equals \(3Y + 60\). Using the fact that all angles sum to 180°:

$$(3Y + 60) + Y + (3Y + 60) = 180$$ $$7Y + 120 = 180$$ $$7Y = 60$$ $$Y \approx 8.57°$$

That's a very small vertex angle, which means the two base angles are quite large (each about 85.7°). It's an unusual triangle, but the math checks out: \(8.57 + 85.71 + 85.71 \approx 180\). ✓

The Relationship Between Sides and Angles

In any triangle, the longest side sits opposite the largest angle, and the shortest side sits opposite the smallest angle. In an isosceles triangle, the two equal sides are opposite the two equal base angles. If the vertex angle is larger than the base angles, the base (the odd side out) is the longest side. If the vertex angle is smaller, the base is the shortest side.