चक्रीय चतुर्भुज

चित्र -1

A quadrilateral ABCD is called cyclic if all the four vertices

of it lie on a circle (see Fig 1).

Here we find that $\angle A+\angle C=180^{\circ }$and $\angle B+\angle D=180^{\circ }$.

Theorems related to Cyclic Quadrilateral are mentioned below.

Theorem 1: The sum of either pair of opposite angles of a cyclic quadrilateral

is $180^{\circ }$.

Theorem 2: If the sum of a pair of opposite angles of a quadrilateral is $180^{\circ }$,

the quadrilateral is cyclic.

उदाहरण

चित्र -2

1: In Fig 2, $ABCD$ is a cyclic quadrilateral in which $AC$ and $BD$ are its diagonals.

If $\angle DBC=55^{\circ }$and $\angle BAC=45^{\circ }$, find $\angle BCD$

Solution:

$\angle CAD=\angle DBC=55^{\circ }$(Angles in the same segment)

Therefore, $\angle DAB=\angle CAD+\angle BAC$

$\angle DAB=55^{\circ }+45^{\circ }=100^{\circ }$

But $\angle DAB+\angle BCD=180^{\circ }$(Opposite angles of a cyclic quadrilateral)

$\angle BCD=180^{\circ }-\angle DAB$

$\angle BCD=180^{\circ }-100^{\circ }=80^{\circ }$