CDS Mathematics · Quadrilaterals

Cyclic Quadrilaterals

A quadrilateral whose vertices lie on one circle; its opposite angles add to 180°, and that rule decides which familiar shapes can be cyclic.

All Quadrilaterals notes CDS Mathematics strategy

Why this matters

Thirteen PYQs, three of them HARD. Most are statement items built on one rule — opposite angles supplementary — applied to parallelograms, trapeziums and angle bisectors. Three use the similar triangles a cyclic quadrilateral makes, across its diagonals or its produced sides.

Concept 1 of 2: Opposite angles and which shapes are cyclic

Opposite vertices look at the same diagonal from the two arcs of the circle, and those two viewing angles always add to

180^\circ

. Any shape whose opposite angles cannot add to

180^\circ

cannot be cyclic.

Definition

Cyclic $\iff$ a pair of opposite angles is supplementary. An exterior angle equals the interior opposite angle.
A cyclic parallelogram is a rectangle; a cyclic trapezium is isosceles; an isosceles trapezium is always cyclic.
Midpoints of a rhombus's sides form a rectangle, so they are always concyclic.
If diagonal $AC$ bisects $\angle C$ , the arcs $AB$ and $AD$ are equal, so $AB = AD$ .
Four sides given in order fix a cyclic quadrilateral, and a right angle makes the opposite diagonal a diameter.

Opposite angles

\angle A + \angle C = \angle B + \angle D = 180^\circ

Worked example

In cyclic

ABCD

\angle A = (3x + 10)^\circ

and

\angle C = (2x + 20)^\circ

. Find

\angle A

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2026 · CDS (II) 2026 — Elementary Mathematics · Q51Moderate

Example 1 · Quadrilaterals · Cyclic Quadrilaterals

In a cyclic quadrilateral ABCD,

\angle A = (4x + 3)^\circ

\angle B = (3y + 9)^\circ

\angle C = (4y - 3)^\circ

\angle D = (5x - 10)^\circ

. What is

\angle A + \angle B

equal to ?

OPPOSITE angles, not adjacent ones

\angle A + \angle C

and

\angle B + \angle D

are

180^\circ

\angle A + \angle B

is not, unless the figure also has parallel sides.

Drill 9 more on opposite angles and which shapes are cyclic

Concept 2 of 2: Similar triangles in a cyclic quadrilateral

Angles in the same segment are equal, so the diagonals cut the figure into two pairs of similar triangles. Produce two sides to meet and the exterior-angle rule makes two more similar triangles.

Definition

Diagonals meeting at $P$ : $\triangle APB \sim \triangle DPC$ , so $\dfrac{[APB]}{[DPC]} = \dfrac{AB^2}{DC^2}$ and $PA \cdot PC = PB \cdot PD$ .
$AB$ and $DC$ produced to meet at $E$ : $\triangle EBC \sim \triangle EDA$ , with $B$ matching $D$ and $C$ matching $A$ .
Match vertices by EQUAL ANGLES, not by the order the letters happen to be written.

Across the diagonals

\dfrac{[APB]}{[DPC]} = \left(\dfrac{AB}{DC}\right)^2

Worked example

The diagonals of cyclic

ABCD

meet at

P

AB = 9

DC = 6

and

[APB] = 27

^2

. Find

[DPC]

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2017 · CDS (II) 2017 — Elementary Mathematics · Q64Moderate

Example 2 · Quadrilaterals · Cyclic Quadrilaterals

The diagonals of a cyclic quadrilateral ABCD intersect at P and the area of the triangle APB is 24 square cm. If AB = 8 cm and CD = 5 cm, then what is the area of the triangle CPD ?

The vertex order in a similarity statement

\triangle EBC

is similar to

\triangle EDA

, not to

\triangle EAD

, because

B

matches

D

. Papers have written the pair in a different order; decide whether a statement means 'these triangles are similar' or 'in this correspondence' before marking it.

Drill 2 more on similar triangles in a cyclic quadrilateral

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Opposite angles and which shapes are cyclic
Opposite angles
$\angle A + \angle C = \angle B + \angle D = 180^\circ$
Similar triangles in a cyclic quadrilateral
Across the diagonals
$\dfrac{[APB]}{[DPC]} = \left(\dfrac{AB}{DC}\right)^2$

Watch out for (2)

OPPOSITE angles, not adjacent ones
→ Opposite angles and which shapes are cyclic
The vertex order in a similarity statement
→ Similar triangles in a cyclic quadrilateral

Test yourself on Quadrilaterals

15 past CDS questions from this chapter, timed at 18 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

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