OPPOSITE ANGLES OF A CYCLIC QUADRILATERAL ARE SUPPLEMENTARY PROOF

Opposite Angles of a Cyclic Quadrilateral are Supplementary Proof :

Here we are going to see the proof of the theorem in cyclic quadrilateral.

Theorem :

Opposite angles of a cyclic quadrilateral are supplementary (or) The sum of opposite angles of a cyclic quadrilateral is 180° Given : O is the centre of circle. ABCD is the cyclic quadrilateral.

To prove : <BAD + <BCD = 180°, <ABC + <ADC = 180°

Construction : Join OB and OD

Proof:

(i) <BAD = (1/2) <BOD

(The angle substended by an arc at the centre is double the angle on the circle.)

(ii) <BCD  =  (1/2) reflex <BOD

(iii) <BAD + <BCD = (1/2) <BOD + (1/2) reflex <BOD

(add (i) and (ii))

<BAD + <BCD  =  (1/2)(<BOD + reflex <BOD)

<BAD + <BCD  =  (1/2) ⋅ (360°)

(Complete angle at the centre is 360°)

<BAD + <BCD = 180°

(iv) Similarly <ABC + <ADC = 180°

Example 1 :

In the figure given below, O is the centre of a circle and <ADC = 120°. Find the value of x. Solution :

ABCD is a cyclic quadrilateral. we have

<ABC + <ADC = 180°

<ABC = 180° - 120°  =  60°

Also <ACB = 90° ( angle on a semi circle )

In triangle ABC we have,

<BAC + <ACB + <ABC  =  180°

<BAC + 90° + 60°  =  180°

<BAC  =  180° - 150°

=  30°

Hence the value of x is 30°.

Example 2 :

In the figure given below, ABCD is a cyclic quadrilateral in which AB || DC. If <BAD  =  100° find

(i) <BCD

(iii) <ABC Solution :

<BAD + <BCD  =  180°

100 + <BCD  =  180

<BCD  =  180 - 100

<BCD  =  80°

<ADC + <ABC  =  180

80 + <ABC  =  180

<ABC  =  180 - 80

<ABC  =  100°

Example 3 :

In the figure given below, ABCD is a cyclic quadrilateral in which <BCD = 100° and <ABD = 50° find <ADB Solution :

<DAB  +  <DCB  =  180°

<DAB  +  100  =  180°

<DAB  =  180 - 100

<DAB  =  80°

<DAB + <ABD + <BDA  =  180

80 + <ABD + 50  =  180

130 + <ABD  =  180

<ABD  =  180 - 130

<ABD  =  50°

Hence the required angle is 50°. After having gone through the stuff given above, we hope that the students would have understood the proof of the theorem in opposite angles of a cyclic quadrilateral

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