Isosceles and Equilateral Triangles

An isosceles triangle is a triangle with at least two sides of equal length. Those two equal sides are called the legs, and the third side is called the base. The two angles that sit opposite the legs, right next to the base, are called the base angles, and the angle between the two legs is the vertex angle. This is one of the key categories used when you study the broader classification of triangles, alongside scalene and equilateral triangles.

An equilateral triangle has all three sides equal in length. Because equal sides always sit opposite equal angles, an equilateral triangle also has three equal angles, and since the interior angles of any triangle add up to \(180^\circ\), each angle in an equilateral triangle measures exactly \(\frac{180^\circ}{3} = 60^\circ\).

The isosceles triangle theorem, often called the base angles theorem, states that if two sides of a triangle are equal, then the angles opposite those sides are also equal. The converse is true as well: if two angles of a triangle are equal, the sides opposite them are equal, which is exactly what makes the triangle isosceles in the first place.

This theorem is useful because if you know one base angle and the vertex angle, you can find every angle in the triangle. For example, if a triangle has a vertex angle of \(40^\circ\) and is isosceles, the two base angles must be equal, so each equals \(\frac{180^\circ - 40^\circ}{2} = 70^\circ\).

Since an equilateral triangle has three equal sides, it automatically satisfies the definition of "at least two equal sides," which makes every equilateral triangle a special case of an isosceles triangle. It is also equiangular, since all three angles equal \(60^\circ\). Not every isosceles triangle is equilateral, but every equilateral triangle is isosceles.

Isosceles triangles can also be classified by their vertex angle:

• Acute isosceles triangle: the vertex angle is less than \(90^\circ\), so all three angles are acute.
• Right isosceles triangle: the vertex angle is exactly \(90^\circ\), and the two base angles are each \(45^\circ\). Because the two legs are equal, you can use the Pythagorean theorem to relate the legs and the hypotenuse directly.
• Obtuse isosceles triangle: the vertex angle is greater than \(90^\circ\), while the two smaller base angles stay equal and acute.

For a right isosceles triangle with legs of length \(a\), the Pythagorean relationship \(a^2 + a^2 = c^2\) simplifies to \(c = a\sqrt{2}\), which is one of the most common applications of the Pythagorean theorem you will see in geometry problems.

To find the area of an isosceles triangle, drop a perpendicular from the vertex angle to the midpoint of the base. This height splits the triangle into two congruent right triangles, so you can use the Pythagorean theorem to find the height if you know the leg length and half the base. Once you have the height \(h\) and base \(b\), the area is:

\(A = \frac{1}{2} \times b \times h\)

Example: An isosceles triangle has legs of length 10 and a base of length 12. Half the base is 6, so the height satisfies \(h^2 + 6^2 = 10^2\), giving \(h^2 = 100 - 36 = 64\), so \(h = 8\). The area is \(A = \frac{1}{2} \times 12 \times 8 = 48\) square units.

An isosceles triangle has a vertex angle of \(52^\circ\). Find the two base angles.

Since the triangle is isosceles, the base angles are equal. Let each base angle equal \(x\). Using the angle sum of a triangle:

\(52^\circ + x + x = 180^\circ\)

\(2x = 128^\circ\)

\(x = 64^\circ\)

Each base angle measures \(64^\circ\), and you can check the answer by adding all three angles: \(52^\circ + 64^\circ + 64^\circ = 180^\circ\).

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