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Question

The angles of a cyclic quadrilateral ABCD are
A=(6x+10),B=(5x)
C=(x+y),D=(3y10)
Find x and y, and hence the values of the four angles.

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Solution

Forming an equation by using the properties of cyclic quadrilateral.​
Given, ABCD is a cyclic Quadrilateral.
Since, the sum of opposite angle of a cyclic quadrilateral measures 180.
Therefore,
A+C=180 and B+D=180
(6x+10)+(x+y)=180
7x+y=18010
7x+y=170 ......(1)
and
5x+(3y10)=180
5x+3y=180+10
5x+3y=190 .....(2)

Finding the values of angles of quadrilateral.
From equation(1), we get
y=1707x .....(3)
Substituting the value from (3) to (2)
5x+3(1707x)=190
5x+51021x=190
16x=190510
16x=320
x=32016
x=20
Substituting the value of x in equation (3) we get,
y=1707(20)
y=170140
y=30
Therefore, values of x and y are 20 are 30 respectively.
Now,
A=6(20)+10=120+10=130
B=5(20)=100
C=20+30=50
D=3(30)10=(9010)=80

Hence the values of x and y are 20 and 30 respectively and the values of the four angles are 130,100,50 and 80.


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