Turn it round

The idea

If two opposite angles of a quadrilateral add up to 180°, then its vertices lie on a circle, so it is a cyclic quadrilateral.

Angles first, circle later

Now turn the last rule round. A quadrilateral ABCD has one pair of opposite angles adding up to 180°. You are not told that it sits on a circle.

A, B and C are fixed. You place D so that ∠B + ∠D = 180°. Where is D, compared with the circle through A, B and C?

• On the circle
• Inside the circle
• Outside the circle

Find 180°

A, B and C are fixed and there is no circle drawn. Touch the shaded region to place D and read ∠B + ∠D. Find places where it is exactly 180°.

The circle appears

If two opposite angles of a quadrilateral add up to 180°, then its vertices lie on a circle: it is a cyclic quadrilateral.

On the circle through A, B and C, the angle at D is exactly 180° − ∠B. A point D inside that circle sees AC at a bigger angle, and a point outside sees it at a smaller angle. So ∠B + ∠D = 180° only when D is on the circle.

Notes

If two opposite angles of a quadrilateral add up to 180°, its vertices are concyclic: it is a cyclic quadrilateral.

Check yourself

In a quadrilateral PQRS, ∠P = 85° and ∠R = 95°. What can we say about PQRS?

Which of these quadrilaterals always has its four vertices on a circle?

In a quadrilateral ABCD, ∠A = 80° and ∠B = 100°. A and B are neighbouring corners. Can we say that ABCD is cyclic?

In a quadrilateral ABCD, ∠B = (3x + 5)⁠° and ∠D = (2x + 15)⁠°. For ABCD to be a cyclic quadrilateral, what must x be?

Answer: 32

(3x + 5) + (2x + 15) = 180, so 5x + 20 = 180 and x = 32.

• It is a parallelogram. In a parallelogram opposite angles are equal. Here ∠P = 85° and ∠R = 95°.
• Its vertices lie on a circle — correct. Yes. ∠P and ∠R are opposite, and 85° + 95° = 180°. So PQRS is a cyclic quadrilateral.
• Nothing can be said. A pair of opposite angles adding up to 180° tells us that the vertices lie on a circle.
• A parallelogram that is not a rectangle. Its opposite angles are equal. They add up to 180° only when each is 90°, and then it is a rectangle.
• A rhombus that is not a square. Its opposite angles are equal. They add up to 180° only when each is 90°, and then it is a square.
• A rectangle — correct. Yes. Each pair of opposite angles is 90° + 90° = 180°.
• No, not from these two angles — correct. Right. The 180° must come from a pair of opposite angles, such as ∠A and ∠C.
• Yes, because 80° + 100° = 180°. A and B are neighbours. The rule is about opposite angles, such as ∠A and ∠C.
• Yes, because every quadrilateral is cyclic. Many quadrilaterals have no circle through all four vertices.