DWG NO. 04.6 — Lesson 6 of 6

Isosceles and Equilateral Triangle Theorems

Unit 4: Congruent Triangles · ~15–30 min

Objective

Apply the Isosceles Triangle Theorem and its converse, along with the equilateral triangle corollaries, to find missing sides and angles.

Base angles of an isosceles triangle

In an isosceles triangle, the two congruent sides are called the legs, the third side is the base, and the two angles touching the base are the base angles.

Isosceles Triangle Theorem: If two sides of a triangle are congruent, then the angles opposite those sides — the base angles — are congruent.
Converse of the Isosceles Triangle Theorem: If two angles of a triangle are congruent, then the sides opposite those angles are congruent.

These two statements let you move freely between "which sides are equal" and "which angles are equal" — whichever one you're given, the theorem hands you the other.

Equilateral triangle corollaries

A triangle is equilateral if and only if it is equiangular.
Each angle of an equilateral triangle measures 60°.

These follow directly from the Isosceles Triangle Theorem: if all three sides are congruent, then all three pairs of base angles must be congruent to each other too — forcing every angle to be equal, and by the Angle Sum Theorem, each must be 180° ÷ 3 = 60°.

Isosceles triangle ABC with legs AB and BC congruent — base angles A and C (marked) are congruent.

Worked Example 1 · Find a base angle

ProblemIsosceles triangle ABC has AB ≅ BC, and angle B = 40°. Find angle A.

1.Since AB ≅ BC, the base angles opposite them — angle C and angle A — are congruent, by the Isosceles Triangle Theorem.

2.Let angle A = angle C = x. By the Angle Sum Theorem: 40 + x + x = 180.

3.2x = 140, so x = 70.

Angle A = 70°

Worked Example 2 · Use the converse to find a side

ProblemTriangle DEF has angle D = angle F = 55°, and side DF = 12. Find side DE.

1.Angle D ≅ angle F, so by the Converse of the Isosceles Triangle Theorem, the sides opposite them are congruent.

2.The side opposite angle D is EF, and the side opposite angle F is DE. So EF ≅ DE — meaning DE and EF are the two legs, and DF = 12 is the base.

3.This tells us DE ≅ EF, but without another numeric side length given, we can only conclude the relationship, not a specific value for DE from this information alone.

DE ≅ EF (both legs of the isosceles triangle); their exact length isn't determined by the given angles alone

Guided practice

1.Isosceles triangle has legs PQ ≅ PR and vertex angle P = 96°. Find each base angle.
Show answer
Base angles Q and R are equal: 180 − 96 = 84, so each base angle = 42°.
2.Triangle has angle X = angle Z = 63°. Which two sides must be congruent?
Show answer
The sides opposite those angles — side XY (opposite Z) and side YZ (opposite X) — must be congruent, by the Converse of the Isosceles Triangle Theorem.
3.An equilateral triangle has one angle labeled 3x. Find x.
Show answer
Every angle in an equilateral triangle is 60°, so 3x = 60, meaning x = 20.
4.Isosceles triangle has base angles of 50° each. Find the vertex angle.
Show answer
180 − 50 − 50 = 80°.
5.A triangle has all three angles equal. What can you conclude about its sides, and what is each angle's measure?
Show answer
All three angles equal means it's equiangular, which by the corollary means it's also equilateral — all three sides are congruent. Each angle measures 60°.

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