#### notes

A quadrilateral ABCD is called cyclic if all the four vertices of it lie on a circle in fig. A quadrilateral is said to be a cyclic quadrilateral, if there is a circle passing through all its four vertices.

`=>`∠A + ∠C = 180°

also ∠B + ∠D = 180°.

#### theorem

**Theorem:** The sum of either pair of opposite angles of a cyclic quadrilateral is 180º. **Given :** A cyclic quadrilateral ABCD . **To Prove :** ∠BAD + ∠BCD = ∠ABC + ∠ADC = 180º. **Construction :** Draw AC and DB.**Proof :** ∠ACB = ∠ ADB and

∠BAC = ∠BDC [Angles in the same segment]

`therefore` ∠ACB + ∠BAC = ∠ADB +∠BDC = ∠ADC

Adding ∠ABC on both the sides, we get

∠ACB + ∠BAC +∠ABC = ∠ADC +∠ABC

But ∠ACB +∠BAC +∠ABC = 180º. [sum of the angles of a triangle]

`therefore` ∠ ADC + ∠ABC =180º.

`therefore` ∠BAD +∠BCD = 360º - (∠ADC + ∠ABC) = 180º.

Hence proved.**Theorem:** If the sum of a pair of opposite angles of a quadrilateral is 180º, the quadrilateral is cyclic.