Angle Relationships • Topic 3 of 3

Angles in a Triangle Sum to 180°

Triangle Angle Sum Theorem

The sum of the three interior angles in ANY triangle is always 180°!

∠A + ∠B + ∠C = 180°

Why is this true? If you tear off the corners of a triangle and put them together, they form a straight line (180°).

Finding Missing Angles:

  • If you know two angles, subtract their sum from 180°
TRIANGLE ANGLE SUM - VISUAL PROOF:

 C
 /
 / 
 / 
 / 
 / 
 A──────────B
 
 ∠A + ∠B + ∠C = 180°


TEAR THE CORNERS:

 Take triangle:
 
 ∠C
 △
 / 
 / 
 ∠A──────∠B
 
 Tear off corners:
 
 ∠A ∠B ∠C
 ┌─┐ ┌─┐ ┌─┐
 │ │ │ │ │ │
 └─┘ └─┘ └─┘
 
 Put together:
 
 ┌─┐┌─┐┌─┐
 │ ││ ││ │
 └─┘└─┘└─┘
 ├─────────┤
 180°


FINDING MISSING ANGLE:

 Triangle angles: 50°, 60°, ?
 
 50° + 60° = 110°
 180° - 110° = 70°
 
 Missing angle = 70°


SPECIAL TRIANGLES:

 Equilateral: 60° + 60° + 60° = 180°
 Right triangle: 90° + 45° + 45° = 180°
 Right triangle: 90° + 30° + 60° = 180°
Solved Examples
1
Worked Example

A triangle has angles 45° and 55°. Find the third angle.

Solution
  • Sum of known = 45° + 55° = 100°
  • Third angle = 180° - 100° = 80°
  • Answer: 80°
2
Worked Example

A right triangle has one angle of 35°. Find the other acute angle.

Solution
  • Right triangle → one angle is 90°
  • Known: 90° + 35° = 125°
  • Third angle = 180° - 125° = 55°
  • Answer: 55°
3
Worked Example

An equilateral triangle has all angles equal. Find each angle.

Solution
  • 3 equal angles = 180°
  • Each angle = 180° ÷ 3 = 60°
  • Answer: 60°
4
Worked Example

A triangle has angles in ratio 2:3:4. Find each angle.

Solution
  • Sum of ratio parts = 2 + 3 + 4 = 9
  • 180° ÷ 9 = 20° per part
  • Angles: 40°, 60°, 80°
  • Answer: 40°, 60°, 80°

Key Points

  • Sum of triangle angles = 180°
  • Equilateral: all 60°
  • Right triangle: 90° + two acute angles (sum 90°)
  • Isosceles: base angles are equal
  • Can find missing angle by subtracting from 180°
Quick Check — 4 MCQs
Tap an option to check your answer0 / 4
Q1.The angles of a triangle sum to:
Explanation: $180^\circ$.
Q2.Each angle of an equilateral triangle is:
Explanation: $60^\circ$.
Q3.If two angles are $50^\circ$ and $60^\circ$, the third is:
Explanation: $180-110$.
Q4.An exterior angle equals the sum of the two:
Explanation: Opposite interior angles.
Practice more MCQs → 📝 Assignment · 35 marks