Isosceles and Equilateral Triangles
Prove and apply the Isosceles Triangle Theorem and its converse: base angles of an isosceles triangle are congruent. Derive that equilateral triangles are equiangular (60° each) and use both theorems in proofs and calculations.
A lesson of Geometry, in Triangles and Congruence, about 10 minutes.
By the end of this lesson, you will be able to apply the Isosceles Triangle Theorem and its converse, solve problems involving isosceles and equilateral triangles, and write proofs using these theorems.
The Big Idea
Isosceles triangles have a built-in symmetry that enforces relationships between their parts. The base angles are congruent — always. This isn't a coincidence but a theorem with a clean proof. Once you know it, many geometry problems that look complex reduce to identifying an isosceles triangle.
Vocabulary
Isosceles triangle: a triangle with at least two congruent sides.
- Legs: the two congruent sides
- Base: the third side (between the base angles)
- Vertex angle: the angle between the two legs
- Base angles: the angles at each end of the base (the non-vertex angles)
Isosceles Triangle Theorem (ITT)
If two sides of a triangle are congruent, then the angles opposite those sides are congruent (the base angles are congruent).
$$\overline{AB} \cong \overline{AC} \implies \angle B \cong \angle C$$
Proof: Draw the angle bisector of $\angle A$, meeting $\overline{BC}$ at $D$.
Then $\triangle ABD \cong \triangle ACD$ by SAS ($\overline{AB} \cong \overline{AC}$ given, $\angle BAD \cong \angle CAD$ by bisector, $\overline{AD} \cong \overline{AD}$ reflexive).
Therefore $\angle B \cong \angle C$ by CPCTC. ∎
Converse of ITT
If two angles of a triangle are congruent, then the sides opposite those angles are congruent.
$$\angle B \cong \angle C \implies \overline{AB} \cong \overline{AC}$$
Equilateral Triangle Theorem
An equilateral triangle is equiangular (all angles = 60°), and an equiangular triangle is equilateral.
Follows directly from the ITT applied twice: if all sides are equal, each pair of base angles is congruent → all three angles equal → 60° each.
Example 1
In $\triangle ABC$, $\overline{AB} \cong \overline{AC}$ and $\angle A = 44°$. Find $\angle B$ and $\angle C$.
Base angles are equal: $\angle B = \angle C$
$$44 + \angle B + \angle C = 180 \implies 44 + 2\angle B = 180 \implies \angle B = 68°$$