Circle theorems: cyclic quadrilaterals

The Maths Habit

Every triangle's corners fit on a circle. Can you draw a four sided shape whose corners do not?

Yes: most of them, such as a long thin kite or a parallelogram that is not a rectangle

Cyclic quadrilateral
A four sided shape with every corner on the circle.

Opposite angles in a cyclic quadrilateral add to 180°

Which angle is opposite A?

Cyclic quadrilateral ABCD C

Could it be cyclic?

Opposite angles of 100° and 90° No

Find x and y.

ABCD75°98°xy
x = 180 − 75 =105°
opposite angles add to 180°
y = 180 − 98 =82°
opposite angles add to 180°

Find x and y.

ABCD110°88°xy
x = 180 − 110 =70°
opposite angles add to 180°
y = 180 − 88 =92°
opposite angles add to 180°

Find x and y.

ABCD75°98°xy
x = 180 − 75 =105°
opposite angles add to 180°
y = 180 − 98 =82°
opposite angles add to 180°

Find x.

ABCDx2x
x + 2x = 180 (opposite angles)
3x = 180
x =60°

Circle theorems: cyclic quadrilaterals worksheet, page 1

Answer sheet

  1. Three angles of a cyclic quadrilateral, in order, are 70°, 95° and 110°. Find the fourth.

    85°

  2. Can every rectangle be drawn with its corners on a circle?

    Yes: opposite angles add to 180°

  1. ABCD is a cyclic quadrilateral. Find angle x.

    x = 106°
    ABCD74°x

  2. ABCD is a cyclic quadrilateral. Find x.

    x = 35
    ABCD3xx + 40

  3. The angles of a quadrilateral, in order, are 70°, 125°, 100° and 65°. Can its corners all lie on a circle? Explain.

    No: 70 + 100 is not 180
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