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Isosceles and equilateral triangle theorems

20 practice questions 2 video lessons Theory + worked examples

Isosceles and Equilateral Triangles

California Geometry • Standard G-CO.10 • Triangle Theorems

Isosceles and Equilateral Triangles is a topic in Triangle Theorems in the California Common Core State Standards. It is aligned to Standard G-CO.10, which requires students to prove theorems about triangles, including the base angles of an isosceles triangle being congruent.

An isosceles triangle has equal base angles opposite its equal sides, and an equilateral triangle is equiangular with three \(60^\circ\) angles.

California Geometry › Triangle Theorems › Isosceles and Equilateral Triangles  —  Standard G-CO.10

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Theory

An isosceles triangle has (at least) two equal sides. The Base Angles Theorem says the angles opposite those equal sides — the base angles — are congruent. The converse also holds: equal base angles force equal sides.

An equilateral triangle has three equal sides; it is also equiangular, with three \(60^\circ\) angles.

Equilateral is a special isosceles triangle — every pair of sides is equal, so every pair of angles is equal.
Isosceles triangle base angles In an isosceles triangle the two base angles opposite the equal sides are congruent. base angles are congruent
Isosceles: the two base angles are congruent.
Equilateral triangle An equilateral triangle has three equal sides and three 60-degree angles. 60° 60° 60° equilateral = equiangular (60° each)
Equilateral: three equal sides and three \(60^\circ\) angles.

The key relationships:

\[\text{isosceles: base angles equal};\qquad \text{equilateral: each angle}=60^\circ\]
isosceles triangles have equal base angles; equilateral triangles have 60-degree angles
Base angle \(=\dfrac{180^\circ-\text{vertex}}{2}\).

How to work with these triangles

  1. Isosceles, given the vertex: base angle \(=\dfrac{180^\circ-\text{vertex}}{2}\).
  2. Isosceles, given a base angle: vertex \(=180^\circ-2(\text{base})\).
  3. Equilateral: every angle is \(60^\circ\).
  4. Converse: equal angles \(\Rightarrow\) equal opposite sides.
Example 1 — Base angles
An isosceles triangle has a vertex angle of \(40^\circ\). Find each base angle.
Solution

The base angles are equal and, with the vertex angle, sum to \(180^\circ\).

\(\dfrac{180^\circ-40^\circ}{2}\)\(=\)\(70^\circ\)
each base angle is 70 degrees
Example 2 — Vertex angle
Each base angle of an isosceles triangle is \(55^\circ\). Find the vertex angle.
Solution

Subtract both base angles from \(180^\circ\).

\(180^\circ-2(55^\circ)\)\(=\)\(70^\circ\)
the vertex angle is 70 degrees
Example 3 — Equilateral angles
Find each angle of an equilateral triangle.
Solution

Equilateral triangles are equiangular; divide \(180^\circ\) by 3.

\(\dfrac{180^\circ}{3}\)\(=\)\(60^\circ\)
each angle is 60 degrees
Example 4 — Converse (equal angles)
A triangle has two \(50^\circ\) angles. What kind of triangle is it?
Solution

By the converse of the base angles theorem, equal base angles mean the sides opposite them are equal, so the triangle is isosceles.

the triangle is isosceles

Common pitfalls

The base angles are the two equal ones, opposite the equal sides — not necessarily the “bottom” angle.
Divide by 2 for base angles, since the two base angles are equal.
Equilateral means equiangular, so you never need to compute — each angle is \(60^\circ\).

Frequently asked questions

What is the base angles theorem?

In an isosceles triangle, the two angles opposite the equal sides (the base angles) are congruent.

What are the angles of an equilateral triangle?

All three are \(60^\circ\), because it is equiangular.

How do you find the base angles from the vertex angle?

Subtract the vertex angle from \(180^\circ\) and divide by 2.

What is the converse of the base angles theorem?

If two angles of a triangle are equal, the sides opposite them are equal, so the triangle is isosceles.

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