Geometry and measuresGCSE Higher

Opposite angles in a cyclic quadrilateral add to 180°

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

  • All four vertices must lie on the circumference. Use opposite angles, not adjacent angles.

The theorem

Opposite angles in a cyclic quadrilateral add to 180 degrees.

Example 1

With numerical values

is a cyclic quadrilateral. Find the angle .

Opposite cyclic angles add to 180 degrees.

Example 2

With algebra

is a cyclic quadrilateral. Find . Each labelled expression is an angle in degrees.

Opposite angles in a cyclic quadrilateral add to 180 degrees.Collect like terms.
Substitute the value of .
Labelled angles
Opposite angles in a cyclic quadrilateral add to 180 degrees.
Collect like terms.
Substitute the value of .
Labelled angles

Your turn

Question 1

Find angle .

Check answer
Opposite cyclic angles add to 180 degrees.
Question 2

Find the value of .

Check answer
Opposite angles in a cyclic quadrilateral add to 180 degrees.Collect like terms.
Substitute the value of .
Labelled angles
Opposite angles in a cyclic quadrilateral add to 180 degrees.
Collect like terms.
Substitute the value of .
Labelled angles
Question 3

Find angle .

Check answer
Opposite cyclic angles add to 180 degrees.
Question 4

Find the value of .

Check answer
Opposite angles in a cyclic quadrilateral add to 180 degrees.Collect like terms.
Substitute the value of .
Labelled angles
Opposite angles in a cyclic quadrilateral add to 180 degrees.
Collect like terms.
Substitute the value of .
Labelled angles
Question 5

Find angle .

Check answer
Opposite cyclic angles add to 180 degrees.
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