A quadrilateral is a 4 sided polygon bounded by 4 finite line segments. The word ‘quadrilateral’ is composed of two Latin words, *quadri* meaning ‘four ‘and *latus* meaning ‘side’. It is a two-dimensional figure having four sides (or edges) and four vertices. A circle is the locus of all points in a plane which are equidistant from a fixed point. If all the four vertices of a quadrilateral ABCD lie on the circumference of the circle then ABCD is a cyclic quadrilateral. In other words, if any four points on the circumference of a circle are joined they form vertices of a cyclic quadrilateral. It can be visualized as a quadrilateral which is inscribed in a circle, i.e. all four vertices of the quadrilateral lie on the circumference of the circle.

### Cyclic Quadrilateral:

In the figure given below, the quadrilateral ABCD is cyclic.

Let us do an activity. Take a circle and choose any 4 points on the circumference of the circle. Join these points to form a quadrilateral. Now measure the angles formed at the vertices of the cyclic quadrilateral. To our surprise, the sum of the angles formed at the vertices is always 360^{o }and the sum of angles formed at the opposite vertices is always supplementary. This property can be stated as a theorem as:

Theorem: In a cyclic quadrilateral, the sum of either pair of opposite angles is supplementary.

Let us now try to prove this theorem.

Given: A cyclic quadrilateral ABCD inscribed in a circle with center O.

Construction: Join the vertices A and C with center O.

The converse of this theorem is also true which states that if opposite angles of a quadrilateral are supplementary then the quadrilateral is cyclic.

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