An isosceles triangle has at least two congruent sides, called the legs. The angle between the legs is the vertex angle; the third side is the base, and the two angles at its ends are the base angles. With the congruence proofs from Proofs with Congruent Triangles we can prove what those base angles do.
Why it works
Isosceles Triangle Theorem. In $\triangle ABC$ with $AB = AC$, draw the bisector of $\angle A$, meeting $\overline{BC}$ at $D$. Then $\overline{AB} \cong \overline{AC}$ (given), $\angle BAD \cong \angle CAD$ (bisector) and $\overline{AD} \cong \overline{AD}$ (shared). So $\triangle ABD \cong \triangle ACD$ by SAS, and by CPCTC $\angle B \cong \angle C$. The converse also holds: if two angles of a triangle are congruent, the sides opposite them are congruent (the same picture with AAS).
Key idea
The angles opposite the congruent sides (the base angles) are congruent, and if two angles are congruent, so are the sides opposite them. An equilateral triangle is equiangular: each angle is $60^\circ$.
Worked example: From the vertex angle
In isosceles $\triangle ABC$ with $AB = AC$, the vertex angle measures $40^\circ$. Find the base angles.
- The base angles are equal and share $180 - 40 = 140^\circ$.
- Each is $140 \div 2 = 70^\circ$.
Answer: $m\angle B = m\angle C = 70^\circ$
Worked example: With algebra
In $\triangle PQR$, $PQ = PR$, $m\angle Q = (3x + 4)^\circ$ and $m\angle R = (5x - 20)^\circ$. Find $m\angle P$.
- $\angle Q$ and $\angle R$ are opposite the equal sides, so $3x + 4 = 5x - 20$, and $x = 12$.
- Each base angle is $3(12) + 4 = 40^\circ$, so $m\angle P = 180 - 80 = 100^\circ$.
Answer: $m\angle P = 100^\circ$
Worked example: The converse
In $\triangle XYZ$, $m\angle X = m\angle Y = 52^\circ$, $XZ = 2n + 3$ and $YZ = 4n - 9$. Find $XZ$.
- Equal angles $X$ and $Y$ mean the sides opposite them are equal: $\overline{YZ}$ (opposite $X$) and $\overline{XZ}$ (opposite $Y$).
- $2n + 3 = 4n - 9$, so $n = 6$ and $XZ = 15$.
Answer: $XZ = 15$
Common mistake
Marking the wrong pair of angles as base angles, especially when the triangle is turned. The base angles are opposite the congruent sides, not next to them; the vertex angle is where the two congruent sides meet. Check: each base angle touches the base and one leg.