Triangles

25. The Triangle's Secret Sum · Why interior angles always add to 180°

Dragging any vertex keeps the interior angles summing to exactly 180°.

∠A = 80°∠A = 80°∠B = 68°∠B = 68°∠C = 32°∠C = 32°ABC
The angle sum theorem says the three interior angles of any triangle always add up to exactly 180°. Drag A, B, or C — the shape changes, each angle changes, but their sum is always the same straight-line value.

What this lesson covers

Try to break it

Try to break it! Pull the vertex until all vertices are in a straight line. Does the sum ever change? No! It's always 180°.

How you build it

Make a triangle ABC.

  • Place vertex A.
  • Place vertex B.
  • Place vertex C (not collinear with A and B).
  • Draw side AB.
  • Draw side BC.
  • Draw side CA. Drag any vertex — the three interior angles always sum to 180°.

The proof, step by step

Prove all angles in a triangle add up to 180°.

  • Draw a line PQ through vertex A such that PQ is parallel to side BC.
  • Since PQ || BC, alternate interior angles are equal: ∠PAB = ∠ABC and ∠QAC = ∠ACB.
  • Angles on a straight line add to 180°: ∠PAB + ∠BAC + ∠QAC = 180°.
  • Substitute the equal angles: ∠ABC + ∠BAC + ∠ACB = 180°. Hence, ∠A + ∠B + ∠C = 180°.

Worked example

In triangle PQR, ∠P = 45° and ∠Q = 85°. Find the measure of ∠R.

The sum of interior angles is 180°. So, ∠R = 180° - (45° + 85°) = 180° - 130° = 50°.

  • 45°
  • 50° — correct
  • 55°
  • 60°
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