Isosceles Triangle Theorem

A bidirectional proof: equal sides imply equal base angles (via SSS congruence), and equal base angles imply equal sides (via AAS congruence).

Keywords: isosceles triangle, base angles, SSS congruence, AAS congruence, congruent triangles

Prerequisites: triangle-congruence · Difficulty: beginner

The isosceles triangle theorem has two parts:

• Forward: If two sides of a triangle are equal, then the angles opposite those sides are equal.
• Converse: If two angles of a triangle are equal, then the sides opposite those angles are equal.

Proof Strategy

Forward: Given $AB = AC$, draw the median $AM$ to the midpoint $M$ of $BC$. By SSS congruence ($AB = AC$, $BM = CM$, $AM = AM$), we get $\triangle ABM \cong \triangle ACM$, so $\angle B = \angle C$.

Converse: Given $\angle B = \angle C$, draw the altitude from $A$ to $BC$. By AAS congruence, $\triangle ABF \cong \triangle ACF$, so $AB = AC$.

The Proof

Step 1: Part 1: Equal Sides → Equal Angles

Given: $AB = AC$ (isosceles triangle). We want to prove $\angle B = \angle C$.

Draw the median $AM$ to the midpoint $M$ of $BC$.

Step 2: $AB = AC \implies \angle B = \angle C$

Claim (Forward): If two sides of a triangle are equal, then the angles opposite those sides are equal: $$AB = AC \implies \angle B = \angle C.$$

Proof sketch. The median $AM$ splits $\triangle ABC$ into $\triangle ABM$ and $\triangle ACM$. By SSS ($AB = AC$, $BM = CM$, $AM = AM$), the two are congruent, so $\angle B = \angle C$ as corresponding angles.

Step 3: SSS Congruence

We prove $\triangle ABM \cong \triangle ACM$ by SSS:

• $AB = AC$ (given — isosceles)
• $BM = CM$ ($M$ is the midpoint of $BC$)
• $AM = AM$ (common side)

Therefore the corresponding angles are equal: $\angle B = \angle C$.

Step 4: Part 2: Equal Angles → Equal Sides

Now for the converse. Given: $\angle B = \angle C$. We want to prove $AB = AC$.

Draw the altitude from $A$ to $BC$, meeting at foot $F$.

Step 5: $\angle B = \angle C \implies AB = AC$

Claim (Converse): If two angles of a triangle are equal, then the sides opposite those angles are equal: $$\angle B = \angle C \implies AB = AC.$$

Proof sketch. The altitude $AF$ splits $\triangle ABC$ into two right triangles $ABF$ and $ACF$. By AAS ($\angle B = \angle C$, both right angles at $F$, $AF$ common), they're congruent, so $AB = AC$.

Step 6: AAS Congruence

We prove $\triangle ABF \cong \triangle ACF$ by AAS:

• $\angle B = \angle C$ (given)
• $\angle AFB = \angle AFC = 90°$ (altitude is perpendicular)
• $AF = AF$ (common side)

Therefore $AB = AC$.

Step 7: The Complete Isosceles Triangle Theorem

We have proven both directions:

AB = AC \iff \angle B = \angle C
    

Equal sides imply equal base angles, and equal base angles imply equal sides. This is the fundamental property of isosceles triangles.

Conclusion

In a triangle, equal sides and equal base angles are equivalent:

AB = AC \iff \angle B = \angle C
    

This bidirectional relationship is one of the most fundamental properties of isosceles triangles.