Theorem

Opposite angles of a cyclic quadrilateral are supplementary

Geometry • Quadrilateral • Cyclic Quadrilateral

Formula

∠ A + ∠ C = ∠ B + ∠ D = 18 0^{\circ}

If the four vertices of a quadrilateral

A B C D

lie on a circle, then its opposite angles are supplementary:

∠ A + ∠ C = 18 0^{\circ}

and

∠ B + ∠ D = 18 0^{\circ}

. The converse also holds — if one pair of opposite angles of a quadrilateral sums to

18 0^{\circ}

, the four vertices are concyclic. An immediate consequence is that the exterior angle at any vertex equals the interior angle at the opposite vertex.

How to use it

1Given one angle of a cyclic quadrilateral, subtract it from $18 0^{\circ}$ to get the angle at the opposite vertex.
2Use the converse to prove four points concyclic: find one pair of opposite angles that sums to $18 0^{\circ}$ and stop there — the second pair then follows from the angle sum $36 0^{\circ}$ .
3Combine it with a shape's own angle properties to force a conclusion: equal opposite angles plus supplementary opposite angles makes each of them $9 0^{\circ}$ .
4The exterior-angle form is usually quicker when a side has been produced: read the interior opposite angle straight off.

Worked example

In a cyclic quadrilateral

P QR S

∠ P = 7 2^{\circ}

and

∠ Q = 9 5^{\circ}

. Find

∠ R

and

∠ S

Step-by-step

1
$∠ R = 18 0^{\circ} - ∠ P = 18 0^{\circ} - 7 2^{\circ} = 10 8^{\circ}$
Why: $∠ P$ and $∠ R$ are opposite angles, so they are supplementary.
2
$∠ S = 18 0^{\circ} - ∠ Q = 18 0^{\circ} - 9 5^{\circ} = 8 5^{\circ}$
Why: The same for the other pair of opposite angles.

Answer

∠ R = 10 8^{\circ}

and

∠ S = 8 5^{\circ}

Watch out

It pairs OPPOSITE angles, not adjacent ones:

∠ A + ∠ B = 18 0^{\circ}

says the sides

A D

and

B C

are parallel, which is a different claim altogether. The converse needs a genuine quadrilateral — four points in order, no three collinear — and supplying both pairs proves nothing extra, since the four angles of any quadrilateral already total

36 0^{\circ}

Builds on

The angle at the centre is twice the angle at the circumference

Also known as

opposite angles of a cyclic quadrilateral are supplementary, cyclic quadrilateral angle theorem, concyclic points condition, exterior angle of a cyclic quadrilateral, condition for four points to lie on a circle

Opposite angles of a cyclic quadrilateral are supplementary

Formula

Worked example

Appears in past papers

2020 (II)

2019 (I)