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# Isosceles Triangle Theorem: Proof, Converse, Examples

[Geometry](/content/tag/geometry/index.html)

TL;DR  
The isosceles triangle theorem says that if two sides of a triangle are equal, then the angles opposite those sides — the base angles — are also equal. Its converse runs the other way: equal base angles force the opposite sides to be equal. This article gives the statement, a full proof using triangle congruence, the converse and its proof, and six worked examples.

BT  
Last updated on June 24, 2026 7 min read

## What Is the Isosceles Triangle Theorem?  
An isosceles triangle is a triangle with (at least) two equal sides. The two equal sides are the **legs**, the third side is the **base**, and the two angles touching the base are the **base angles**.  
  
The **isosceles triangle theorem** states:

> If two sides of a triangle are congruent, then the angles opposite those sides are congruent.

In a triangle ABC with AB=AC, the theorem guarantees ∠B=∠C. The two base angles are [congruent angles](/content/math/geometry/congruent-angles/index.html) — equal in measure. This is the angle counterpart to the equal-side definition: matching sides produce matching opposite angles. It places the isosceles triangle firmly among the [types of triangle](/content/math/geometry/types-of-triangle/index.html) defined by their symmetry.

## Proof of the Isosceles Triangle Theorem  
**Given:** Triangle ABC with AB=AC. **To prove:** ∠B=∠C.  
**Construction:** Draw AD, the angle bisector of ∠A, meeting BC at D.

Now compare triangles ABD and ACD, step by step:

AB=AC(given)  
∠BAD=∠CAD(AD bisects ∠A)  
AD=AD(common side)  
By the **SAS** (Side-Angle-Side) rule, triangle ABD≅ triangle ACD. This is one of the standard rules in the [triangle congruence theorem](/content/math/geometry/triangle-congruence-theorem/index.html) set.

Since corresponding parts of congruent triangles are congruent (CPCT):

t∠B=∠C  
That completes the proof. The base angles are equal.

## The Converse of the Isosceles Triangle Theorem  
The converse swaps what is given and what is proved:

> If two angles of a triangle are congruent, then the sides opposite those angles are congruent.

So in triangle ABC, if ∠B=∠C, then AB=AC. The converse is also true — and that two-way street is what makes the theorem so useful: equal sides tell you equal angles, and equal angles tell you equal sides.

## Proof of the Converse  
**Given:** Triangle ABC with ∠B=∠C. **To prove:** AB=AC.  
**Construction:** Draw AD, the angle bisector of ∠A, meeting BC at D.

Compare triangles ABD and ACD:

∠B=∠C(given)  
∠BAD=∠CAD(AD bisects ∠A)  
AD=AD(common side)  
By the **AAS** (Angle-Angle-Side) rule, triangle ABD≅ triangle ACD.

By CPCT:

AB=AC  
The opposite sides are equal. (Notice the proof uses AAS here, whereas the forward theorem used SAS — the given information changed, so the matching congruence rule changed with it.)

## Examples of Isosceles Triangle Theorem  
### Example 1  
**In triangle ABC, AB=AC and ∠B=50°. Find ∠C.**  
By the isosceles triangle theorem, the base angles opposite the equal sides are equal:  
∠C=∠B=50°.  
**Final answer:** 50°.

### Example 2 (a tempting shortcut that fails)  
**In triangle PQR, PQ=PR and the apex angle ∠P=40°. A student claims each base angle is 40° too. Find the base angles.**  
_Wrong attempt._ The reflex is "isosceles means two equal angles, and the apex is 40°, so the base angles are 40° each." That treats the apex angle as one of the equal pair.  
_Why it breaks._ The equal angles are the _base_ angles (opposite the equal sides), not the apex angle between the equal sides.  
Three 40° angles would total only 120°, not 180°.  
_Correct._ The base angles are equal; call each x. Use the angle sum:  
40°+x+x=180°  
2x=140°  
x=70°.  
**Final answer:** Each base angle is 70° — the apex angle is the _odd one out_, not part of the equal pair.

### Example 3  
**In triangle XYZ, XY=XZ and ∠X=80°. Find the base angles.**  
The base angles are equal. With apex ∠X=80°:  
∠Y+∠Z=180°−80°=100°  
∠Y=∠Z=100°/2=50°.  
**Final answer:** Each base angle is 50°.

### Example 4 (using the converse)  
**In triangle ABC, ∠B=∠C=65°, and side AB=9 cm. Find side AC.**  
By the converse, equal base angles force the opposite sides equal:  
AC=AB=9 cm.  
**Final answer:** 9 cm.

### Example 5  
**In an isosceles triangle, one base angle is (2x+10)° and the other is (3x−5)°. Find x.**  
The base angles are equal:  
2x+10=3x−5  
10+5=3x−2x  
x=15.  
**Final answer:** x=15 (each base angle is then 40°).

### Example 6  
**A triangle has angles 70°, 70°, and 40°. Is it isosceles, and which sides are equal?**  
Two angles are equal (70° each), so by the converse the triangle is isosceles. The equal sides are the ones _opposite_ the two 70° angles.  
**Final answer:** Yes, it is isosceles; the two sides opposite the 70° angles are equal.

## Why the Isosceles Triangle Theorem Matters  
> "Equal sides cannot help but produce equal angles."  
The WHY is symmetry made rigorous. Long before coordinates or trigonometry, Euclid needed a way to _prove_ that a balanced shape really is balanced — that intuition about symmetry could be trusted as a theorem, not just a feeling. The isosceles triangle theorem ( **Proposition 5, Book I of Euclid's _Elements_**) is that bridge, and almost every later [geometrical proof](/content/math/geometry/geometrical-proofs/index.html) about symmetric figures leans on it.  
Where it earns its keep:  
- **Structural symmetry.** A symmetric A-frame, a gabled roof, a suspension-bridge tower — wherever two members of equal length meet a base, the equal base angles are guaranteed, so engineers can predict load angles from lengths alone.  
- **Constructions and bisectors.** The theorem is the engine behind compass-and-straightedge constructions of perpendicular bisectors and angle bisectors; the isosceles triangle is the scaffold those constructions quietly build.  
- **Trigonometry's foundations.** Splitting an isosceles triangle by its axis of symmetry yields two congruent right triangles — the same move that powers the special-angle values and the law of sines.

## Where Students Trip Up on the Isosceles Triangle Theorem  
### Mistake 1: Treating the apex angle as one of the equal base angles  
**Where it slips in:** Finding angles when only the apex angle is given.  
**Don't do this:** Assume the apex angle equals each base angle.  
**The correct way:** The equal angles are the **base angles**, opposite the equal sides. The apex angle (between the equal sides) is usually different.  
### Mistake 2: Confusing the theorem with its converse  
**Where it slips in:** Choosing what you are allowed to conclude.  
**Don't do this:** Use the theorem (sides → angles) when you were given angles, or vice versa.  
**The correct way:** Equal _sides_ let you conclude equal _angles_ (the theorem). Equal _angles_ let you conclude equal _sides_ (the converse). Match the direction to what you were given.  
### Mistake 3: Picking the wrong sides as the equal pair  
**Where it slips in:** Applying the converse from equal angles.  
**Don't do this:** Make the side _between_ the two equal angles the equal side.  
**The correct way:** The equal sides are **opposite** the equal angles. The side between the two equal angles is the base — and the base is the odd side out.

## Key Takeaways  
- The **isosceles triangle theorem** states that equal sides produce equal opposite (base) angles.  
- Its **converse** states that equal base angles produce equal opposite sides — both directions are true.  
- The proof drops the angle bisector from the apex, giving two triangles congruent by SAS, then uses CPCT.  
- The base angles are opposite the equal sides; the apex angle (between them) is usually different.  
- The most common error is treating the apex angle as one of the equal pair, or confusing the theorem with its converse.