Find an angle
A standard method from Basic properties of circles: the steps and worked examples.
Steps
- Mark the centre, radii, and any right angles on the diagram.
- Radii are equal, so look for isosceles triangles.
- Look for one of the "big five" shapes:
| Shape you see | Fact | Reason (short form) |
|---|---|---|
| an angle at O and one at the circumference on the same arc | centre = 2 × circumference | ∠ at centre twice ∠ at ⊙ce |
| two angles standing on the same arc | equal | ∠s in the same segment |
| a diameter with a triangle on it | 90° | ∠ in semi-circle |
| four points on the circle | opposite angles add to 180° | opp. ∠s, cyclic quad. |
| a tangent touching the circle | 90° to the radius | tangent ⊥ radius |
- Write one new angle per line, with the reason in brackets.
Reason short forms differ slightly between textbooks. Use the student's own book's version.
Worked examples
Example 1: A circle has radius 10 cm. Chord AB = 16 cm. Find the distance from the centre O to AB.
Let M be the foot of the perpendicular from O to AB.
AM = 16 ÷ 2 = 8 cm (line from centre ⊥ chord bisects chord)
OM = √(10² − 8²) = 6 cm (Pythagoras' theorem)Example 2: ABCD is a cyclic quadrilateral with ∠A = 2x and ∠C = x + 30°. Find ∠A.
2x + (x + 30°) = 180° (opp. ∠s, cyclic quad.)
3x = 150°
x = 50°
∠A = 100°Example 3: In a circle with centre O, ∠AOB = 130°. C is on the major arc. Find ∠ACB.
∠ACB = 130° ÷ 2 = 65° (∠ at centre twice ∠ at ⊙ce).
For the whole topic, read Basic properties of circles: key points, common mistakes and how the DSE tests it.