Geometry and measuresGCSE Higher

Combining circle theorems with reasons

Recognise the theorem, follow two examples, then try five questions.

  • Name the theorem at each step. is the centre and , , and lie on the circle.

The theorem

Use one theorem to find an intermediate angle, then use a second theorem to find the required angle.

Example 1

With numerical values

is the centre and is cyclic. Find the angle .

The angle at the centre is twice the angle at the circumference.Opposite angles in a cyclic quadrilateral add to 180 degrees.
The angle at the centre is twice the angle at the circumference.Opposite angles in a cyclic quadrilateral add to 180 degrees.

Example 2

With algebra

is the centre and is tangent at . Find . Give a reason for each step.

The alternate segment theorem.The angle at the centre is twice the angle at the circumference.
Substitute the value of .
Labelled angles
The alternate segment theorem.
The angle at the centre is twice the angle at the circumference.
Substitute the value of .
Labelled angles

Remember
Give a reason for every angle you find.

Your turn

Question 1

Find angle .

Check answer
The alternate segment theorem.The angle at the centre is twice the angle at the circumference.
The alternate segment theorem.The angle at the centre is twice the angle at the circumference.
Question 2

Find the value of .

Check answer
The angle at the centre is twice the angle at the circumference.Halve the centre angle; the opposite cyclic angles then add to 180 degrees.Collect like terms.
Substitute the value of .
Labelled angles
The angle at the centre is twice the angle at the circumference.
Halve the centre angle; the opposite cyclic angles then add to 180 degrees.
Collect like terms.
Substitute the value of .
Labelled angles
Question 3

Find angle .

Check answer
The angle at the centre is twice the angle at the circumference.Opposite angles in a cyclic quadrilateral add to 180 degrees.
The angle at the centre is twice the angle at the circumference.Opposite angles in a cyclic quadrilateral add to 180 degrees.
Question 4

Find the value of .

Check answer
The alternate segment theorem.The angle at the centre is twice the angle at the circumference.
Substitute the value of .
Labelled angles
The alternate segment theorem.
The angle at the centre is twice the angle at the circumference.
Substitute the value of .
Labelled angles
Question 5

Find angle .

Check answer
The alternate segment theorem.The angle at the centre is twice the angle at the circumference.
The alternate segment theorem.The angle at the centre is twice the angle at the circumference.
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