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Question

ABCD is a cyclic quadrilateral. Find the angles of the cyclic quadrilateral.


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Solution

Finding the angles of cyclic quadrilateral:

Step 1:Finding the values of x and y,

In cyclic quadrilateral, the sum of the opposite angles is 1800.

From figure, we get,

A=4y+20B=3y-5C=-4xD=-7x+5

Therefore,C+A=180

4y+204x=1804x+4y=160

yx=40 …………………….(1)

And, B+D=180

3y-57x+5=180

3y-7x=180 ……………….(2)

On multiplying 3 to equation (1), we get,

3y3x=120……………………………(3)

On subtracting equation (2) from equation (3), we get

3y-3x-(3y-7x)=120-1803y-3x-3y+7x=-604x=-60x=-15

Substituting this value in equation (1), we get,

yx=40y-(-15)=40y+15=40y=25

Step 2: Finding the angles :

On substituting the values of x and y, we get,

A=4y+20=4×25+20=120B=3y-5=3×25-5=70C=-4x=-4×(-15)=60D=-7x+5=-7×(-15)+5=110

Hence, the angles of the cyclic quadrilateral are A=120,B=70,C=60,D=110.


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