Isosceles Triangle Theorem
In short: The isosceles triangle theorem says that when two sides of a triangle are congruent, the two angles opposite those sides are congruent too.
GeometryGeometryG-CO.C.10
What is Isosceles Triangle Theorem?
The isosceles triangle theorem links equal sides to equal angles. Take the two matching sides, the legs, and look across the triangle: the corners facing them, the base angles, have the same measure.
The converse is true as well. If two angles of a triangle are equal, the sides facing them are equal. Combined with the angle sum of \(180^{\circ}\), the theorem gives a quick formula: a triangle whose apex measures \(a\) has base angles of \(\frac{180 – a}{2}\) each.
How to spot it
- A triangle is labeled isosceles, or two of its sides carry matching tick marks.
- Two angles of a triangle are given as equal and a missing side is requested.
- A roof truss, a kite or a pennant has two equal edges and a measured tip angle.
- An equation sets two angle expressions equal because the sides opposite them are equal; that is the isosceles triangle theorem at work.
The method
- Find the two congruent sides and the two angles that face them.
- Set the facing angles equal, since equal sides give equal angles.
- Use \(180^{\circ}\) for the whole triangle to write one equation for the unknown.
- Solve, then check that the three angles total \(180^{\circ}\).
Isosceles Triangle Theorem: solved example
Problem
A gable on a barn is an isosceles triangle with a 44° angle at the peak. Find the measure of each base angle.
Solution
The two sloping sides are equal, so the angles opposite them are equal. Call each one \(b\). The angles total \(180^{\circ}\): \(44 + b + b = 180\), so \(2b = 136\).
Therefore \(b = 68\). Check: \(44 + 68 + 68 = 180\). Each base angle measures \(68^{\circ}\).
Try it
Question 1
The base angles of an isosceles triangle are each \((2x + 9)^{\circ}\) and the peak angle is \(62^{\circ}\). Find \(x\).
Show the answer
\(2(2x + 9) + 62 = 180\), so \(4x + 80 = 180\) and \(x = 25\). Each base angle is \(59^{\circ}\).
Question 2
In \(\triangle JKL\), \(\angle J = \angle K = 55^{\circ}\) and \(JL = 12\). Find \(KL\).
Show the answer
By the converse, the sides opposite equal angles are equal. \(KL\) faces \(\angle J\) and \(JL\) faces \(\angle K\), so \(KL = 12\).
Isosceles Triangle Theorem vs. Triangle Angle Sum Theorem
The angle sum theorem holds for every triangle and says the three angles total \(180^{\circ}\). The isosceles triangle theorem needs two equal sides and adds new information: two of the angles are equal. Together they solve problems like the barn gable: 44, 68 and 68.
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