Isosceles Triangle Theorem (Base Angles)
If two sides of a triangle are congruent, then the angles opposite those sides are also congruent. This is the Base Angles Theorem — and its converse is equally powerful.
Introduction
In an isosceles triangle, the two equal sides are called legs, and the third side is the base. The angles at the base are the base angles, and the angle between the two legs is the vertex angle. This lesson proves the deep connection between equal sides and equal angles.
Past Knowledge
Triangle classification (5.1.1). Angle Sum Theorem (5.1.2). Congruent triangles intro.
Today's Goal
Prove and apply the Base Angles Theorem and its converse.
Future Success
Equilateral triangles (5.1.5), congruence proofs (5.2), and coordinate proofs.
Key Concepts
Isosceles Triangle Theorem (Base Angles Theorem)
If two sides of a triangle are congruent, then the angles opposite those sides are congruent.
If , then .
Converse of the Base Angles Theorem
If two angles of a triangle are congruent, then the sides opposite those angles are congruent.
If , then .
Isosceles Triangle Vocabulary
| Part | Definition |
|---|---|
| Legs | The two congruent sides |
| Base | The third side (between the base angles) |
| Vertex angle | The angle between the two legs (opposite the base) |
| Base angles | The two congruent angles at the ends of the base |
Theorem & Proof
Two-Column Proof: Isosceles Triangle Theorem
Given: with
Prove:
Strategy: Reflect across the perpendicular bisector of and use the fact that reflections preserve angle measure (Chapter 4).
Click each step to follow the reflection proof. Congruent legs have tick marks. Line m is the perpendicular bisector. Base angles are preserved by reflection.
| # | Statement | Reason |
|---|---|---|
| 1 | Given | |
| 2 | Let be the perpendicular bisector of | Construction (every segment has a perpendicular bisector) |
| 3 | Point lies on line | Any point equidistant from and lies on the perpendicular bisector of (since ) |
| 4 | Reflect across line : , , | A reflection across swaps and (they are symmetric about ); is on so it maps to itself |
| 5 | The reflection maps to | The image of under the reflection is (vertices map ) |
| 6 | Reflections are rigid motions and preserve angle measure (Chapter 4) |
∎ If two sides of a triangle are congruent, the base angles are congruent.
Worked Examples
Finding a Base Angle
In isosceles with , the vertex angle . Find the base angles.
By the Base Angles Theorem, . Let .
Using the Converse
In , and . What can you conclude about the sides?
Since , by the Converse of the Base Angles Theorem, the sides opposite these angles are congruent:
(side opposite = side opposite )
The third angle:
— the triangle is isosceles with .
Algebraic Side Lengths
In isosceles with , the base angles each measure and the vertex angle measures . Find , all angles, and determine if the legs have length , what is that length?
Angles:
Base angles: each
Vertex angle:
Check: ✓
Leg length:
. Base angles = 68°, vertex = 44°, legs = 71 units.
Common Pitfalls
Mixing Up “Opposite” Sides and Angles
The base angles are opposite the legs (congruent sides), not adjacent to them. Side is opposite — the angle at the vertex not on that side.
Assuming All Isosceles Triangles Are Acute
A vertex angle can be obtuse (e.g., 100°, with two 40° base angles). It can also be a right angle (90°, giving two 45° base angles). The base angles are always acute, but the vertex angle can be anything < 180°.
Forgetting the Converse Exists
The converse is just as useful: equal angles → equal opposite sides. If a problem gives you two equal angles, you can immediately conclude the triangle is isosceles.
Real-Life Applications
Structural Engineering — A-Frame Roofs
A-frame buildings use isosceles triangles so that both sides of the roof bear the same load. The equal side lengths guarantee equal base angles, which means symmetric weight distribution — critical for structural stability.
Optics — Prism Light Paths
Isosceles prisms use the base angles theorem to predict how light refracts symmetrically. Since the two faces form equal angles with the base, incoming light exits at a predictable, symmetric angle — essential for spectrometers and binoculars.
Practice Quiz
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