Maths Essentials, Unit 25, Lesson 2 of 5

The circle theorems

GCSE Maths, Circles and circle theorems, 25 minutes

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The angle facts for points on a circle, and the exact reason to write next to each one.

What you will learn

  • Use the angle at the centre and the angle in a semicircle
  • Use angles in the same segment and cyclic quadrilaterals
  • Use the perpendicular from the centre to a chord
  • Write the right theorem as a reason

Four angle facts

For points A, B, C and D on a circle with centre O:

  • The angle at the centre is twice the angle at the circumference when both stand on the same arc.
  • The angle in a semicircle is 90°: if AB is a diameter, angle ACB = 90°. (The angle at the centre is then 180°, and half of it is 90°.)
  • Angles in the same segment are equal: angles ACB and ADB, on the same side of the chord AB, are the same size.
  • Opposite angles of a cyclic quadrilateral add up to 180°. A cyclic quadrilateral has all four corners on the circle.

In a cyclic quadrilateral ABCD, angle A is opposite angle C and angle B is opposite angle D. With A = 95° and B = 80°: C = 180 − 95 = 85° and D = 180 − 80 = 100°. Check: the four add up to 360°.

If the angle at the circumference is obtuse, the angle at the centre on the same arc is reflex: angle ACB = 110° gives reflex angle AOB = 2 × 110 = 220°, standing on the major (longer) arc AB.

O120°60°ABC

The rest of this lesson and its 7 practice questions are locked.

In this lesson

  1. Four angle facts
  2. Two more facts you will need
  3. Say which theorem
  4. Try it: Move the points

7 practice questions, marked as you go; about 25 minutes in all.

In this unit: Circles and circle theorems

The parts of a circle, the circle theorems and how to give them as reasons, tangents, chains of reasoning, and the equation of a circle centred at the origin with its tangents.

All of Circles and circle theorems

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