# Find the value of x , y , z , w from the figure, where O is the centre of the circle, angle AOC = 110 and angle OAB = 65? Dear Student,

Given : $\angle$ AOC  =  150$°$  and $\angle$ OAB  =  65$°$

We know " The central angle subtended by two points on a circle is twice the inscribed angle subtended by those points . "

So,

$\angle$ AOC  = 2 $\angle$ ADC  , Substitute given value we get

110$°$ = 2 z  ,

z  =  55$°$

And

$\angle$ AOC  + y  = 360$°$                     (  Center angles )

110$°$ + y  = 360$°$

y    = 250$°$

And we use same theorem as we use to find value of ' x '  and get

y  = 2 $\angle$ ABC , Now we substitute above value and get

2 x  =  250$°$

x  = 125$°$

From angle sum property of quadrilateral we get in quadrilateral OABC  :

$\angle$ AOC  + $\angle$ OAB  + $\angle$  ABC  +  $\angle$ OBC  =  360$°$  , Substitute all values we get

110$°$ + 65$°$  + 125$°$ + w  = 360$°$ ,

300$°$ + w = 360$°$ ,

w  = 60$°$

Therefore,

x  = 125$°$  , y  = 250$°$  , z = 55$°$ and w = 60$°$                                                                   ( Ans )

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Regards

• -1
110
• 0
x=110(angle subtended by an arc at the centre is exactly half of the angle subtended by it on the remaining part of                 the circle)
since angle O is a complete circle it has an angle of 360 degrees
angle AOC =250 degrees
Similarly angle ADC = half of 250 degrees
angle ADC = 125 degrees = z

• 1
x = 125
y = 250
z = 55
w = 60
• 2
w??
• 1
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