Base angles of the isosceles triangle

A triangle with two congruent sides is called isosceles. Its most celebrated property — already attributed to Thales — concerns the angles opposite those two sides.

Theorem — Base angles of the isosceles triangle

If a triangle has two congruent sides ($AC\cong BC$), then the angles opposite those sides are congruent: $AC\cong BC \implies \widehat{CAB}\cong\widehat{CBA}$

Proof

The strategy consists of creating two pairs of congruent triangles, exploiting symmetric extensions of the sides.

1. Construction. Extend $CA$ beyond $A$ by a segment $AD$; extend $CB$ beyond $B$ by a segment $BE\cong AD$. Join $D$ to $B$ and $E$ to $A$.
2. I consider. The triangles $CBD$ and $CAE$: $BC\cong AC$ (hypothesis), $DC = AC+AD \cong BC+BE = CE$ (sums of congruents), $\widehat{C}$ in common.
3. I deduce. By the first criterion: $\triangle CBD\cong\triangle CAE$, therefore $BD\cong AE$ and $\widehat{CDB}\cong\widehat{CEA}$.
4. I consider. Let us now move on to the triangles $ABD$ and $BAE$: $AB$ in common, $AD\cong BE$ (construction), $BD\cong AE$ (just proved).
5. I deduce. By the third criterion: $\triangle ABD\cong\triangle BAE$, therefore $\widehat{BAD}\cong\widehat{ABE}$.
6. Verified. Finally, $\widehat{CAB}$ and $\widehat{BAD}$ are supplementary, as are $\widehat{CBA}$ and $\widehat{ABE}$. Being supplementary to congruent angles, we conclude $\widehat{CAB}\cong\widehat{CBA}$.

$\blacksquare$

Links

Topics: Euclidean geometry
Concepts: Angle · Congruence criteria · Proof · Isosceles triangle
Skills: Proving · Synthetic geometry