What is the angle in a semicircle?

90°. The angle standing on a diameter is always a right angle.

The angle at the centre is 140°. What is the angle at the circumference on the same arc?

70°. Half the angle at the centre.

What do the opposite angles of a cyclic quadrilateral add up to?

180°. All four corners lie on the circle.

Two tangents are drawn to a circle from the same point. What is true of them?

They are equal in length. From the point to where each touches the circle.

Circle theorems · Cambridge IGCSE Mathematics (9

In plain words

Draw lines inside a circle and the angles they make obey some surprising rules. There are only a few, and every circle question is a matter of spotting which one is hiding in the diagram.

Three things to know

The angle in a semicircle is 90°. A tangent meets a radius at 90°. The angle at the centre is twice the angle at the circumference standing on the same arc.
Angles in the same segment (standing on the same arc) are equal. Opposite angles of a cyclic quadrilateral add up to 180°.
Two tangents drawn from the same point are equal in length. The angle between a tangent and a chord equals the angle in the alternate segment.

Worked example

A, B and C are points on a circle with centre O. The angle AOB at the centre is 110°. Find the angle ACB at the circumference.

The angle at the centre is twice the angle at the circumference on the same arc.
Angle ACB = 110 ÷ 2.
= 55°.

Tips and tricks

Look for radii: any two make an isosceles triangle, with two equal angles.
Quote the theorem in words as your reason. That is where the marks are.

5 questions, about 2 minutes.

It lands in your notebook with its questions as flashcards.

Circle theorems: 5 questions and answers

These are the quiz’s questions. Do the quiz first, then come back here for the ones that got you.

What is the angle in a semicircle?
- 45°
- 90° (the answer)
- 180°
- 60°
The angle standing on a diameter is always a right angle.
What angle does a tangent make with the radius at the point where it touches?
- 45°
- 60°
- 90° (the answer)
- 180°
They are perpendicular.
The angle at the centre is 140°. What is the angle at the circumference on the same arc?
- 40°
- 70° (the answer)
- 140°
- 280°
Half the angle at the centre.
What do the opposite angles of a cyclic quadrilateral add up to?
- 90°
- 180° (the answer)
- 270°
- 360°
All four corners lie on the circle.
Two tangents are drawn to a circle from the same point. What is true of them?
- They are perpendicular.
- They are equal in length. (the answer)
- They are parallel.
- One is twice the other.
From the point to where each touches the circle.

5 questions

Tap an answer and you’ll see straight away whether it’s right, and why.

Worksheet

3 questions, 6 marks. Write your answers on paper, then check them.

Circle theorems

Cambridge IGCSE Mathematics (9–1) 0980 · 6 marks · papermunch.org

Name ______________________________ Date ______________

AB is a diameter of a circle and C is a point on the circle. Angle CAB = 35°. Find angle CBA.[2]
Show answerHide answer
55°. Angle ACB is 90° (angle in a semicircle); 180 − 90 − 35.
I got this one
ABCD is a cyclic quadrilateral with angle A = 80°. Find angle C, giving a reason.[2]
Show answerHide answer
100°. Opposite angles of a cyclic quadrilateral add up to 180°.
I got this one
The angle at the circumference standing on an arc is 40°. Find the angle at the centre standing on the same arc.[2]
Show answerHide answer
80°.
I got this one

Answers: Circle theorems

1. 55°. Angle ACB is 90° (angle in a semicircle); 180 − 90 − 35.
2. 100°. Opposite angles of a cyclic quadrilateral add up to 180°.
3. 80°.