205. The Triangle's Secret Sum · Angles always add up to 180°

Try to break it

Drag A, B, or C anywhere. The three angle readouts shift, but they always add to exactly 180°. Stretch the triangle until it's almost a flat line — the angles still sum to 180°. Try to find a position where they don't; impossible.

How you build it

Construct triangle ABC.

• Place point A as the apex of the triangle.
• Place point B as the second vertex of the triangle.
• Place point C as the third vertex of the triangle.
• Draw segment AB.
• Draw segment BC as the base of the triangle.
• Draw segment CA to complete triangle ABC.

The proof, step by step

Prove that the angles of a triangle add up to 180°.

• Draw a line through A parallel to BC.
• ∠B = ∠DAB (Alternate interior angles)
• ∠C = ∠EAC (Alternate interior angles)
• ∠DAB + ∠BAC + ∠EAC = 180° (Angles on a straight line)
• Therefore, ∠B + ∠A + ∠C = 180°

Worked example

In a triangle, if two angles measure 50° and 70°, what is the measure of the third angle?

The sum of angles in a triangle is 180°. So, 180° - (50° + 70°) = 180° - 120° = 60°.

• 40°
• 50°
• 60° — correct
• 70°