Opposite corners

The idea

The sum of either pair of opposite angles of a cyclic quadrilateral is 180°.

Opposite angles

ABCD is a quadrilateral with all four corners on a circle. The corners A and C are opposite each other, and so are B and D. Look at the angle at A and the angle at the opposite corner C.

What do you think ∠A + ∠C comes to, in a cyclic quadrilateral?

• Always 180°
• Always 90°
• It changes every time

Add them up

Drag the corners round the circle. Read the angles at the opposite corners. Try 3 very different quadrilaterals.

Half of a full turn

The sum of either pair of opposite angles of a cyclic quadrilateral is 180°.

∠A is half of the angle that arc BCD makes at the centre. ∠C is half of the angle that the other arc BAD makes at the centre. The two arcs together make a full turn, 360°. So ∠A + ∠C = ½ × 360° = 180°. The same works for ∠B + ∠D.

Notes

In a cyclic quadrilateral, each pair of opposite angles adds up to 180°.

Check yourself

ABCD is a cyclic quadrilateral and ∠A = 70°. What is ∠C, in degrees?

Answer: 110 °

Opposite angles of a cyclic quadrilateral add up to 180°, so ∠C = 180° − 70° = 110°.

ABCD is a cyclic quadrilateral. Which pair of angles always adds up to 180°?

ABCD is a cyclic quadrilateral with ∠A = 100° and ∠B = 80°. What is ∠D, in degrees?

Answer: 100 °

∠D is opposite ∠B, so ∠D = 180° − 80° = 100°.

In cyclic quadrilateral ABCD, the opposite angles are ∠A = (2x + 10)⁠° and ∠C = (3x + 20)⁠°. Find x.

Answer: 30

(2x + 10) + (3x + 20) = 180, so 5x + 30 = 180 and x = 30.

• ∠A and ∠B. A and B are neighbouring corners. Their sum changes as the corners move.
• ∠A and ∠C — correct. Yes. A and C are opposite corners.
• All four angles together. The four angles of any quadrilateral add up to 360°. Each pair of opposite angles makes 180°.