Triangles

30. The 60° Secret · Why equilateral triangles always hide three equal angles

When XY = YZ = XZ, all interior angles are exactly 60°.

XZ✓ EQUILATERALXY = 360XY = 360YZ = 360YZ = 360XZ = 360XZ = 360∠X = 60°∠X = 60°∠Y = 60°∠Y = 60°∠Z = 60°∠Z = 60°Y
An equilateral triangle is one where all three sides are equal — and as a consequence all three interior angles are 60°. The triangle on the canvas is NOT locked to that shape; drag vertex Y and watch the side lengths and angle measures change. The ✓ EQUILATERAL badge only appears when XY = YZ = XZ — try to find that special configuration where every angle becomes exactly 60°.

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Selina ICSE: Triangles

What this lesson covers

Try to break it

Drag point Y up and down. Watch how the sides and angles change. When does it become equilateral again?

How you build it

Construct an equilateral triangle.

  • Place point A — one end of the side AB.
  • Place point B a little way from A — this fixes the side length AB.
  • Draw side AB (Segment tool: click A, then B).
  • Open the compass to length AB: click on A for the centre, then click exactly on B to set the radius. An arc swings out around A.
  • The radius is now locked to AB. Click once on B — the same arc swings out around B. Where the two arcs cross is point C.
  • Mark point C at a spot where the two arcs cross — either crossing works (above or below line AB). Your click snaps onto the exact intersection, so AC and BC both come out equal to AB.
  • Draw side AC (Segment tool: click A, then C).
  • Draw side BC (Segment tool: click B, then C). Triangle ABC is equilateral — all three sides equal AB by construction.

The proof, step by step

Prove that every angle of an equilateral triangle is 60°.

  • Given: In △XYZ, XY = YZ = XZ (All sides are equal).
  • In any triangle, angles opposite to equal sides are equal.
  • Therefore, ∠X = ∠Y = ∠Z.
  • We know that the sum of interior angles in a triangle is always 180°.
  • So, 3 × ∠X = 180° → ∠X = 60°. Hence, all angles are 60°.

Worked example

In an equilateral triangle PQR, if side PQ = 10 cm, what is the measure of ∠R?

In an equilateral triangle, all three sides are equal, which means all three interior angles are also equal. Since the sum of angles in any triangle is 180°, each angle measures 180° ÷ 3 = 60°. Therefore, ∠R = 60°.

  • 30°
  • 45°
  • 60° — correct
  • 90°
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