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

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

Strategy: for the forward direction, drop the median $AM$ to the midpoint $M$ of $BC$ and use SSS congruence to get $\angle B=\angle C$. For the converse, drop the altitude from $A$ to foot $F$ and use AAS congruence to get $AB=AC$.

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: $AB=AC⟹\angle B=\angle C$

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

Proof sketch. The median $AM$ splits $△ABC$ into $△ABM$ and $△ACM$. 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 $△ABM\cong △ACM$ 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: $\angle B=\angle C⟹AB=AC$

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

Proof sketch. The altitude $AF$ splits $△ABC$ 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 $△ABF\cong △ACF$ 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:

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

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

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