Geometry and measuresGCSE Higher / IGCSE Extended

An exterior angle of a cyclic quadrilateral equals the opposite interior angle

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

  • Extend one side in a straight line. This follows from the opposite-angle theorem.

The theorem

An exterior angle of a cyclic quadrilateral equals the opposite interior angle.

Example 1

With numerical values

is cyclic and is extended through to . Find the angle .

The exterior angle equals the opposite interior angle.

Example 2

With algebra

is cyclic and is extended through to . Find . Each labelled expression is an angle in degrees.

The exterior angle equals the opposite interior angle.Collect like terms.
Substitute the value of .
Labelled angles

Your turn

Question 1

Find angle .

Check answer
The exterior angle equals the opposite interior angle.
Question 2

Find the value of .

Check answer
The exterior angle equals the opposite interior angle.Collect like terms.
Substitute the value of .
Labelled angles
Question 3

Find angle .

Check answer
The exterior angle equals the opposite interior angle.
Question 4

Find the value of .

Check answer
The exterior angle equals the opposite interior angle.Collect like terms.
Substitute the value of .
Labelled angles
Question 5

Find angle .

Check answer
The exterior angle equals the opposite interior angle.
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