Question

In the given fig. ABCD is a cyclic quadrilateral and O is the centre of circle. If angle CBQ = 60°and x = 2y, find the valies of x, y and z. Also find angle AOC.

Answer 1

given, quadrilateral ABCD is a cyclic quadrilateral. where <CBQ = 60°

<ABC + <CBQ = 180°(LP)
<ABC+60°=180°
<ABC=180°-60°
<ABC= 120°

<ABC+<ADC= 180°( sum of opp angles of cyclic quadrilateral)
120°+<ADC=180°
<ADC=180°-120°
<ADC= 60°= z
2<z = <AOC ( angle subtended by arc ........theorem )
2*60°=<AOC
120°= <AOC

<BCQ= 180°-60°-y....( 1 )
<BCD = 120°-y
<ABP+<ABC=180°
120-<ABP =180
<ABP=180-120
<ABP=60°
<BAP=180-<ABP-2y ( given, x=2y )........(2)
<BAD = 60°-2y

ABCD is a cyclic quadrilateral. so,
<BAD+<BCD=120-3y=180°

y=20° & x=40°

Answer 2

In ABCD is cyclic quadrilateral.  So ∠ABC + z = 180°
At B, ∠ABC = 180 - 60°  = 120°    
       =>  z = 60°

As Arc ABC makes ∠AOC at O and ∠ z at D on the circle,   
     ∠AOC = 2 z = 120°

In ΔBCQ,  ∠BCQ = 180° - y - 60° = 120° - y
Hence, at C, ∠BCD = 180° - ∠BCQ = 60° + y

In ΔAPB,  ∠PBA = 180° - ∠ABC = 60°
                 ∠PAB = 180° - x - 60° = 120° - 2 y
    Hence,  ∠DAB = 180 - ∠PAB = 60° + 2y

In the cyclic quadrilateral ABCD,  ∠DAB + ∠BCD = 120° + 3 y  = 180°

Hence  y = 20°   and  x = 40°