Angle Sum of a Triangle

Learn that the angles in any triangle always add up to 180°

The Angle Sum Property

Key Rule:

The sum of angles in a triangle = 180°

$\angle A + \angle B + \angle C = 180°$

This property is true for ALL triangles - equilateral, isosceles, scalene, acute-angled, right-angled or obtuse-angled. Let's check it with different types:

Example 1: Acute-angled Triangle

Acute-angled triangle: 50° + 70° + 60° = 180°

$50° + 70° + 60° = 180°$ ✓

Example 2: Right-angled Triangle

Right-angled triangle: 90° + 40° + 50° = 180°

$90° + 40° + 50° = 180°$ ✓

In a right-angled triangle, the two other angles always add up to 90°

Example 3: Obtuse-angled Triangle

Obtuse-angled triangle: 115° + 25° + 40° = 180°

$115° + 25° + 40° = 180°$ ✓

Example 4: Equilateral Triangle

Equilateral triangle: 60° + 60° + 60° = 180°

$60° + 60° + 60° = 180°$ ✓

Each angle in an equilateral triangle = 180° ÷ 3 = 60°

Finding Unknown Angles

If we know two angles in a triangle, we can find the third angle:

Add the two known angles
Subtract from 180°

Worked Example 1: Find angle x

Find angle x

Solution:

Sum of angles in a triangle = 180°

$75° + 45° = 120°$

$x = 180° - 120°$

$x = 60°$

Worked Example 2: Isosceles Triangle

In isosceles triangle ABC, AB = AC and angle A = 40°. Find angles B and C.

Isosceles triangle with AB = AC

Solution:

Step 1: Since AB = AC, the base angles are equal

$\angle B = \angle C$

Step 2: Use angle sum = 180°

$\angle B + \angle C = 180° - 40° = 140°$

$\angle B = 140° \div 2 = 70°$ (since ∠B = ∠C)

$\angle B = \angle C = 70°$

Special Cases to Remember

In a Right-angled Triangle:

The two acute angles add up to 90°

If one angle = 90°, then:

$\text{Sum of the other two angles} = 180° - 90° = 90°$

In an Equilateral Triangle:

All angles equal 60°

All 3 angles are equal:

$\text{Each angle} = 180° \div 3 = 60°$

Practice Problems

Practice 1: Find angle a

Find angle a

ShowHide solution

$80° + 55° = 135°$

$a = 180° - 135°$

$a = 45°$

Practice 2: Find angle y in a right-angled triangle

Right-angled triangle: Find angle y

ShowHide solution

$y = 180° - 90° - 28°$

$y = 62°$

(Quick check: In a right-angled triangle, the other two angles add up to 90°: 62° + 28° = 90° ✓)

Practice 3: Isosceles triangle with equal base angles

In isosceles triangle PQR, PQ = PR. If angle Q = 65°, find angles P and R.

Isosceles triangle with PQ = PR

ShowHide solution

Step 1: Since PQ = PR, base angles Q and R are equal

$\angle R = \angle Q = 65°$

Step 2: Find angle P

$\angle P = 180° - 65° - 65° = 50°$

$\angle P = 50°$ , $\angle R = 65°$

Practice 4: Right-angled isosceles triangle

Triangle DEF is a right-angled isosceles triangle with the right angle at F. Find angles D and E.

Right-angled isosceles triangle with DF = EF

ShowHide solution

Step 1: Isosceles means two sides are equal. The right angle cannot be one of two equal angles (two right angles would already make 180°), so the equal sides are DF and EF, as marked.

The angles opposite equal sides are equal: $\angle D = \angle E$

Step 2: Use angle sum = 180°

$\angle D + \angle E = 180° - 90° = 90°$

Since ∠D = ∠E:

$\angle D = 90° \div 2 = 45°$

$\angle D = \angle E = 45°$

Key Takeaways

Sum of angles in ANY triangle = 180°
To find an unknown angle: add known angles, subtract from 180°
In a right-angled triangle: the two other angles add up to 90°
In an equilateral triangle: each angle = 60°
In an isosceles triangle: base angles are equal