How to Understand the Properties of Isosceles and Equilateral Triangles

An isosceles triangle has two equal sides, and the angles opposite those sides are equal too. The base angles theorem and its converse work in both directions, so equal angles also prove equal sides.

• The angles opposite the equal sides are congruent.
• The altitude drawn to the base bisects both the base and the vertex angle.

• All interior angles are congruent and each measures \(60^\circ\).
• All altitudes (or heights) are congruent.
• Every equilateral triangle is also equiangular.

Worked examples: isosceles and equilateral triangles

Practice questions on isosceles and equilateral triangles

1. Calculate the area and perimeter of an isosceles triangle with a base of \(14 \text{ cm} \) and legs of \(15 \text{ cm} \) each.
2. Determine the height of an equilateral triangle with a side length of \(9 \text{ cm} \).

1. Using the Pythagoras’ theorem, height \(h = \sqrt{15^2 – 7^2} = \sqrt{164} \approx 12.8 \text{ cm} \). Area \(= \frac{1}{2} \times 14 \times 12.8 = 89.6 \text{ cm}^2 \). Perimeter \(= 14 + 2 \times 15 = 44 \text{ cm} \).
2. Using the area formula relationship, height \(h = \sqrt{9^2 – 4.5^2} = \sqrt{56.25} = 7.5 \text{ cm} \).

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