Angles at a Point

Learn that angles around a point always add up to 360°

Understanding Angles at a Point

When several lines meet at a single point, they form angles around that point. These angles always add up to 360° because together they make one complete turn.

Key Rule:

Sum of angles at a point = 360°

Example: Four Angles at a Point

a + b + c + d = 360°

Angles at a point sum to 360°

∠a + ∠b + ∠c + ∠d = 60° + 70° + 130° + 100° = 360°

$\angle a + \angle b + \angle c + \angle d = 60° + 70° + 130° + 100° = 360°$

Finding Unknown Angles at a Point

To find an unknown angle at a point:

Add all the known angles
Subtract from 360°

Worked Example: Find the unknown angle

Three angles meet at a point. Find ∠h.

Solution:

Angles at a point sum to 360°

$110° + 90° = 200°$

$\angle h = 360° - 200° = 160°$

Equal Angles at a Point

Sometimes angles at a point are equal. Think of a wind turbine with 3 equally spaced blades!

Example: Wind Turbine with 3 Equally Spaced Blades

a + b + c = 360°

Angles at a point sum to 360°

Three equal angles: each is 360° ÷ 3 = 120°

How to find each angle:

If 3 angles are equal and sum to 360°:

$\text{Each angle} = 360° \div 3 = 120°$

Practice Problems

Practice 1: Find angle x

Three angles meet at a point. Find ∠x.

ShowHide solution

$135° + 25° = 160°$

$\angle x = 360° - 160° = 200°$

Practice 2: Find angle d

Four angles meet at a point. Find ∠d.

ShowHide solution

$95° + 75° + 110° = 280°$

$\angle d = 360° - 280° = 80°$

Practice 3: Four equal angles

Four equal angles meet at a point. What is the size of each angle?

ShowHide solution

If 4 angles are equal and sum to 360°:

$\text{Each angle} = 360° \div 4 = 90°$

Key Takeaways

Angles at a point always add up to 360°
This is because a full turn around a point is 360°
To find an unknown angle: add known angles, then subtract from 360°
When all the angles at a point are equal: each angle = 360° ÷ the number of angles