Triangles
95. The 180° Rule · Every triangle's angles add up to a straight line
The sum of the interior angles of any triangle is always 180°.
The Angle Sum Property: the three interior angles of any triangle always add up to exactly 180°. This holds for every triangle — equilateral, isosceles, scalene, right, acute, obtuse.
What this lesson covers
Try to break it
Try dragging C so close to line AB that the triangle flattens. What happens to the angles?
How you build it
Make a triangle with a line through A parallel to BC.
- Place point A, the first non-collinear vertex.
- Place point B.
- Place point C so the three points are not on one straight line.
- Draw the side from A to B.
- Draw the side from B to C.
- Draw the side from C to A to complete triangle ABC.
The proof, step by step
Prove that the interior 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° (straight line)
- Therefore, ∠B + ∠A + ∠C = 180°
Worked example
In ΔABC, ∠A = 55° and ∠B = 60°. Find the measure of ∠C.
We know that the sum of interior angles of a triangle is 180°. So, ∠A + ∠B + ∠C = 180°. Substituting the given values: 55° + 60° + ∠C = 180° ⇒ 115° + ∠C = 180° ⇒ ∠C = 180° - 115° = 65°.
- 55°
- 60°
- 65° — correct
- 70°