Fundamental Concepts
17. Angles Around a Point · The sum is always 360°
The four angles around O always add up to 360°.
When several rays share the same vertex, they split the full circle around that vertex into a set of angles. These angles around a point always sum to 360° (a complete revolution).
What this lesson covers
Try to break it
Try to make the angles add up to something other than 360°. You can't! Angles around a point will always add up to 360°
How you build it
Draw four rays from a point.
- Place point O — the centre where all rays meet.
- Place point A.
- Draw ray OA.
- Place point B in a new direction.
- Draw ray OB.
- Place point C in another direction.
- Draw ray OC.
- Place point D in the last open direction.
- Draw ray OD. The four angles around O wrap a full turn — they sum to 360°.
The proof, step by step
Prove that all angles with vertex O add up to 360°.
- The angles ∠AOB, ∠BOC, ∠COD, and ∠DOA are adjacent angles around point O.
- Together, these angles completely cover the space around point O.
- The total angle around a single point is defined as 360°.
- Therefore, ∠AOB + ∠BOC + ∠COD + ∠DOA = 360°.
Worked example
In the figure, four angles are formed around point O. If ∠AOB = 85°, ∠BOC = 110°, and ∠COD = 75°, what is the measure of ∠DOA?
The sum of all angles around a point is 360°. Subtract the given angles from 360°: 360° - (85° + 110° + 75°) = 360° - 270° = 90°.
- 90° — correct
- 95°
- 100°
- 105°