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,

Please find below the solution to the asked query:

Given : AOC  =  150°  and OAB  =  65°

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

So,

AOC  = 2 ADC  , Substitute given value we get

110° = 2 z  ,

z  =  55°    

And

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 ABC , Now we substitute above value and get

2 x  =  250°

x  = 125°

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

AOC  + OAB  +   ABC  +  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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110
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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
 
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x = 125
y = 250
z = 55
w = 60
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w??
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