# In a cyclic quadrilateral ABCD, /_ A= (2x+4)° , /_=(y+3)°, /_ C= (2y+10)° and /_D=(4x-5)°.find the measure of each angle.

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by minal61

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by minal61

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As it is a cyclic quadrilateral../_A+/_C=/_B+/_D=180......2x+4+2y+10=180.....2x+2y+14=180...,,x+y+7=90..... x+y=83..eq1.....Now for eq2..y+3+4x-5=180....4x+y-2=180...4x+y=182..y=182-4x NOW SUBSTITUTING THE VALUE OF y..in eq1 we have..x+182-4x=83...-3x=-99...x=33....Again in eq1..x+y=83....33+y=83....y=50..(Ans)

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In a cyclic quadrilateral, Opposite angles are supplementary.

angle A + angle C = 180 deg. and also angle B + angle D = 180 deg.

Now 2x + 4 + 2y + 10 = 180 => x+y = 83 deg.

similarly, y+3+4x-5 = 180 => 4x+y = 182 deg.

solving the two simultaneous equations in x and y we get,

3x = 99 deg; x= 33 deg

y = 50 deg.

now calculate A,B,C and D.

A = 70 deg => C = 110 deg

B = 53 deg => D = 127 deg

angle A + angle C = 180 deg. and also angle B + angle D = 180 deg.

Now 2x + 4 + 2y + 10 = 180 => x+y = 83 deg.

similarly, y+3+4x-5 = 180 => 4x+y = 182 deg.

solving the two simultaneous equations in x and y we get,

3x = 99 deg; x= 33 deg

y = 50 deg.

now calculate A,B,C and D.

A = 70 deg => C = 110 deg

B = 53 deg => D = 127 deg