Geometry / Unit 5: Congruent Triangles
5.6 Isosceles and Equilateral Triangles
Notes
An isosceles triangle has at least two congruent sides, the legs. The third side is the base, and the angles touching the base are base angles.
- Isosceles Triangle Theorem: if two sides are congruent, the angles opposite them are congruent.
- Converse: if two angles are congruent, the sides opposite them are congruent.
- An equilateral triangle is also equiangular: each angle is $60^\circ$.
- Example: the vertex angle is $40^\circ$, so each base angle is $\displaystyle \frac{180 - 40}{2} = 70^\circ$.
Practice
An isosceles triangle has a vertex angle of $50^\circ$. What is each base angle?
$\displaystyle \frac{180 - 50}{2} = 65$.
In $\triangle ABC$, $\angle A \cong \angle C$, $AB = 2x + 3$ and $BC = 4x - 7$. Find $AB$.
Sides opposite congruent angles are congruent: $2x + 3 = 4x - 7$, so $x = 5$ and $AB = 13$.
A base angle of an isosceles triangle measures $72^\circ$. What is the vertex angle?
$180 - 2(72) = 36$.
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