Geometry and measuresGCSE Higher

The angle at the centre is twice the angle at the circumference standing on the same arc

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

  • Use the same endpoints on the circle. The circumference vertex lies on the other arc.

The theorem

The angle at the centre is twice the angle at the circumference standing on the same arc.

Example 1

With numerical values

is the centre. Find the angle .

The angle at the centre is twice the angle at the circumference.

Example 2

With algebra

is the centre. Find . Each labelled expression is an angle in degrees.

The centre angle is twice the circumference angle.Collect like terms.
Labelled anglesSubstitute the value of .

Your turn

Question 1

Find angle .

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

Find the value of .

Check answer
The centre angle is twice the circumference angle.Collect like terms.
Labelled anglesSubstitute the value of .
The centre angle is twice the circumference angle.
Collect like terms.
Labelled anglesSubstitute the value of .
Question 3

Find angle .

Check answer
The angle at the centre is twice the angle at the circumference.
Question 4

Find the value of .

Check answer
The centre angle is twice the circumference angle.Collect like terms.
Labelled anglesSubstitute the value of .
Question 5

Find angle .

Check answer
The angle at the centre is twice the angle at the circumference.
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