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: 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.
Theorem: If the sum of a pair of opposite angles of a quadrilateral is 180º, the quadrilateral is cyclic.
Shaalaa.com | Cyclic Quadrilaterals
In the given figure, A, B and C are three points on a circle with centre O such that ∠BOC = 30° and ∠AOB = 60°. If D is a point on the circle other than the arc ABC, find ∠ADC.
Prove that the line of centres of two intersecting circles subtends equal angles at the two points of intersection.
If diagonals of a cyclic quadrilateral are diameters of the circle through the vertices of the quadrilateral, prove that it is a rectangle.
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A chord of a circle is equal to the radius of the circle. Find the angle subtended by the chord at a point on the minor arc and also at a point on the major arc.