Isosceles & Equilateral Triangles
Master every property, formula, and SAT strategy for these high-frequency triangle types.
SAT Tip: Isosceles and equilateral triangles appear in roughly 15–20% of SAT geometry questions. Knowing their properties cold saves you precious time.
1. Introduction — Triangle Classification by Sides
| Type | Equal Sides | Key Fact |
|---|---|---|
| Scalene | 0 | No sides equal; no angles equal |
| Isosceles | ≥ 2 | At least two congruent sides → two congruent angles |
| Equilateral | 3 | All sides equal → all angles 60° |
Key Relationship: Every equilateral triangle is isosceles (it has at least two equal sides). But not every isosceles triangle is equilateral.
Interactive visualization — available in the full lesson
2. Parts of an Isosceles Triangle
Interactive visualization — available in the full lesson
Vocabulary You Must Know
| Legs | The two congruent sides (red in diagram): $AB = AC$ |
| Base | The third side $BC$ — the "unequal" side |
| Vertex Angle | $\angle A$ — the angle between the two legs (opposite the base) |
| Base Angles | $\angle B$ and $\angle C$ — the angles adjacent to the base |
| Altitude to Base | Segment from $A$ perpendicular to $BC$. In an isosceles triangle this segment is simultaneously the altitude, median, angle bisector, and perpendicular bisector of the base. |
| Line of Symmetry | The altitude from the vertex angle is the axis of symmetry. |
Standard Notation: If $AB = AC$, then $BC$ is the base, $\angle A$ is the vertex angle, and $\angle B = \angle C$ are the base angles.