Overview

Equilateral and isosceles triangles provide symmetry. Use equal sides to convert length information into angle information quickly, and create special right triangles by dropping altitudes.

Key Ideas

• Equilateral: all sides equal, all angles $60^\circ$.
• Isosceles: equal legs imply equal base angles.
• The altitude from the apex of an isosceles triangle also bisects the base.
• In an equilateral triangle, any altitude is also a median and angle bisector.

Core Skills

Mark Equal Parts

Once two sides or two angles are equal, label the corresponding parts and apply the triangle angle sum.

Drop an Altitude

In an isosceles or equilateral triangle, dropping an altitude creates two congruent right triangles with helpful ratios.

Use Symmetry

If a line is a median, angle bisector, or altitude in a symmetric triangle, it often serves as the axis of symmetry for the entire figure.

Worked Example

In isosceles triangle $ABC$ with $AB=AC$, the apex angle $\angle A$ is $36^\circ$. Find each base angle.

Each base angle is $(180^\circ-36^\circ)/2 = 72^\circ$.

More Examples

Example 1: Equilateral Altitude

An equilateral triangle has side $10$. Find its altitude.

Split into a 30-60-90 triangle. The altitude is $5\sqrt{3}$.

Example 2: Isosceles Angle Chase

Triangle $ABC$ is isosceles with $AB=AC$. If $\angle B = 50^\circ$, find $\angle A$.

$\angle B = \angle C = 50^\circ$, so $\angle A = 180^\circ - 100^\circ = 80^\circ$.

Strategy Checklist

• Identify the equal sides or equal angles first.
• Use triangle angle sum early to reduce variables.
• Drop an altitude to create a special right triangle if lengths are needed.
• Check that the altitude lands on the base, not the extension.

Common Pitfalls

• Applying isosceles angle facts when the equal sides are not identified.
• Forgetting to split the remaining angle sum in half.
• Assuming the altitude always bisects the base in a non-isosceles triangle.

Practice Problems