Theorems on Cyclic Quadrilateral

In this section we will discuss theorems on cyclic quadrilateral.

A quadrilateral is called Cyclic quadrilateral if its all vertices lie on the circle.
It has some special properties which other quadrilaterals, in general, need not have. Here we have proved some theorems on cyclic quadrilateral.

1) The opposite angles of a Cyclic - quadrilateral are supplementary.
Given : A cyclic quadrilateral ABCD.
Prove that : ∠A + ∠C = 180° ang ∠B + ∠D = 180°
Construction : Join AC and BD.

Statements	Reasons
1) ∠ACB = ∠ADB	1) Angles in the same segment.
2) ∠BAC = ∠BDC	2) Angles in the same segment
3)∠ACB + ∠BAC = ∠ADB + ∠BDC	3) Addition property
4) ∠ACB + ∠BAC = ∠ADC	4) Add ∠ABC on both sides.
5) ∠ABC + ∠ACB + ∠BAC = ∠ABC + ∠ADC	5) From Above.
6) 180o = ∠ABC + ∠ADC	6) Sum of the angle of a triangle is 180o
7) ∠B + ∠D = 180o	7) Opposite angles of cyclic quadrilateral.
8) ∠A + ∠B + ∠C + ∠D	8) Measure of a quadrilateral.
9) ∠A + ∠C = 360o - (∠B + ∠D)	9) From Above.
10) ∠A + ∠C = 360o - 180o = 180o	10) Angle sum property
11) ∠A + ∠C = 180o and ∠B + ∠D = 180o	11) From above .So opposite angles are supplementary.

2) If one side of a cyclic quadrilateral is produced, then the exterior angle is equal to the interior opposite angle.
Given : A cyclic quadrilateral ABCD one of whose side AB is produced to E.
Prove that : ∠CBE = ∠ADC

Statements	Reasons
1) ∠ABC + ∠ADC = 180o	1) Opposite angles of cyclic quadrilateral
2) ∠ABC + ∠CBE = 180o	2) Linear Pair angles.
3) ∠ABC + ∠ADC = ∠ABC + ∠CBE	3) From above.
4) ∠ADC = ∠CBE	4) subtraction property

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Theorems on Circle

• Theorems on Chord
• Theorems on Chord and Subtended Angle
• Theorems on Arc and Angle
• Theorems on Cyclic Quadrilateral

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