Key takeaway: The isosceles triangle theorem states that if two sides of a triangle are congruent, then the angles opposite those sides are congruent, so an isosceles triangle has equal base angles.
Grade 10 – GeometryG-CO.C.10Geometry and Measurement
Definition of Isosceles Triangle Theorem
The isosceles triangle theorem links equal sides to equal angles. In a triangle with two equal sides, called legs, the two angles at the other end of those legs, the base angles, have the same measure.
The converse also holds: a triangle with two equal angles has two equal sides, opposite those angles. Together they let you move between side facts and angle facts. The line that bisects the apex angle also cuts the base in half at a right angle, which is how the theorem is usually proved.
Formula and notation
The isosceles triangle theorem in symbols: if \(AB = AC\) in \(\triangle ABC\), then
\[ \angle B \cong \angle C \]
The angle at the apex \(A\) satisfies \(m\angle A = 180^{\circ} – 2\,m\angle B\). Tick marks show equal sides, and arcs show equal angles.
Key facts
- The equal angles are always the ones opposite the equal sides.
- Knowing the apex angle gives each base angle as half of 180 minus the apex.
- An equilateral triangle is a special case with all three angles equal to 60 degrees.
- The converse lets equal angles prove that two sides are equal.
Isosceles Triangle Theorem example
Apply the isosceles triangle theorem: in \(\triangle ABC\), \(AB = AC\). The base angles measure \(m\angle B = (3x + 5)^{\circ}\) and \(m\angle C = (5x – 19)^{\circ}\). Find \(x\) and the apex angle \(\angle A\).
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Equal legs give equal base angles: \(3x + 5 = 5x – 19\).
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Then \(24 = 2x\), so \(x = 12\), and each base angle is \(3(12) + 5 = 41^{\circ}\).
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The angles of a triangle sum to \(180^{\circ}\): \(m\angle A = 180 – 41 – 41\).
\(x = 12\) and the apex angle is \(98^{\circ}\).
Common mistake
Watch out
In a triangle with \(AB = AC\), Wen says \(\angle A\) and \(\angle B\) must be equal, since they touch the equal side \(AB\).
Why it is wrong
The equal angles are the ones opposite the equal sides. Side \(AB\) is opposite \(\angle C\) and side \(AC\) is opposite \(\angle B\), so \(\angle B \cong \angle C\). Touching a side is not the same as facing it.
Related terms and lessons
Browse the glossary: A–Z list · Geometry vocabulary · Geometry and Measurement