Circles: Cyclic Properties — Part 2: Cyclic quadrilaterals: visual lesson, ICSE Class 10 Mathematics

Lesson

Circles: cyclic quadrilaterals, Part 2

When all four corners of a quadrilateral lie on a circle, its opposite angles add up to $180^\circ$ . In this part we prove that, use it, and use its converse to show that four points lie on a circle.

In this lesson you will

Prove that opposite angles of a cyclic quadrilateral are supplementary
Use the exterior angle of a cyclic quadrilateral equals the interior opposite angle
Combine circle properties with triangles and algebra to find angles
Use the converse to show that four points are concyclic

Cyclic $ABCD$: $75^\circ + 105^\circ = 180^\circ$ and $78^\circ + 102^\circ = 180^\circ$.

Concept

What you will be able to do

Prove that opposite angles are supplementary
Use the exterior angle property
Combine circle facts with triangles and algebra
Use the converse: show that points are concyclic

Part 2 on one card.
Property	Statement	Typical use
Opposite angles	$\angle A + \angle C = 180^\circ$ , $\angle B + \angle D = 180^\circ$	find a missing angle of a cyclic quadrilateral
Exterior angle	equals the interior opposite angle	find an angle without the supplement
Converse	supplementary opposite angles: the corners are concyclic	prove that four points lie on a circle
Cyclic parallelogram	is a rectangle	conclude right angles

Concept

Cyclic quadrilaterals

A quadrilateral is cyclic when all four vertices lie on one circle; the vertices are then concyclic. A circle through $A$ , $B$ , $C$ , $D$ is called the circumcircle.

not every quadrilateral is cyclic
rectangles and squares always are

$ABCD$ is a cyclic quadrilateral: its vertices are concyclic.

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