Triangles
98. The Equilateral Promise · every corner locked at 60°
If all sides are equal, then all interior angles are exactly 60°.
Equilateral triangle properties: all three sides are equal AND all three angles are 60° (since 180° / 3 = 60°). The two conditions are equivalent — equal sides imply equal angles and vice versa.
What this lesson covers
Try to break it
Now try to break the pattern. Drag a vertex out of place. As soon as the sides are no longer equal, the angles change from 60°. The rule is strict: equal sides force equal angles.
How you build it
Construct an equilateral triangle.
- Draw a line segment AB of any length — this is the first side.
- Set the compass to length AB, place the needle at A, and draw an arc above the line.
- Keep the same width, place the needle at B, and draw another arc to cross the first one.
- Mark point C where the two arcs cross.
- Draw the side from A to C.
- Draw the side from B to C to complete the equilateral triangle.
The proof, step by step
Prove that an equilateral triangle has three 60° angles.
- Given: In ΔABC, AB = BC = CA.
- Since AB = AC, ∠C = ∠A. (Angles opposite equal sides are equal)
- Since AB = BC, ∠C = ∠B. Thus, ∠A = ∠B = ∠C.
- Angle sum property: ∠A + ∠B + ∠C = 180°.
- Substitute: 3∠A = 180° ⇒ ∠A = 60°. Hence, all angles are 60°.
Worked example
In an equilateral triangle ABC, if the perimeter is 36 cm, what is the measure of each interior angle?
By definition, all angles in an equilateral triangle are equal. Since the sum of angles in any triangle is 180°, each angle must be 180° ÷ 3 = 60°. The side length or perimeter does not change the angle measure.
- 45°
- 60° — correct
- 90°
- 120°