The angles of a cyclic quadrilateral ABCD are `angleA=(6x+10)^(@), angleB=(5x)^(@), angle C =(x+y)^(@)` and `angle D = (3y-10)^(@)`.

Explanation

We know that, by property of cyclic quadrilateral,
Sum of opposite angles `= 180^(@) " " `[`:' angleA=(6x+10)^(@), angleC=(x+y)^(@)`, given]
`rArr " " 4x+y=170 " " ...(i)`
and `" " angleB+angleD=(5x)^(@)+(3y-10)^(@)=180^(@) " " ` [`:' angleB=(5x)^(@), angleD=(3y-10)^(@)`, given]
`rArr " " 5x+3y=190^(@) " " ...(ii)`
On multiplying Eq. (i) by 3 and then subtracting, we get
`3 xx (7x +y)-(5x+3y)=510^(@)-190^(@)`
`rArr " "21x+3y-5x-3y=320^(@)`
`rArr " " 16x=320^(@)`
`:. " " x=20^(@)`
On putting `x=20^(@)` in Eq. (i), we get
`7xx20+y=170^(@)`
`rArr " " y=170^(@)-140^(@) rArr y=30^(@)`
`:. " " angle=(6x+10)^(@)=6xx20^(@)+10^(@)`
`" " =120^(@)+10^(@)=130^(@)`
`" " angleB=(5x)^(@)=5xx20^(@)=100^(@)`
`" " angleC=(x+y)^(@)=20^(@)+30^(@)=50^(@)`
`" " angleD=(3y-10)^(@)=3xx30^(@)-10^(@)`
`" " =90^(@)-10^(@)=80^(@)`
Hence, the required values of x and y are `20^(@)` and `30^(@)` respectively and the values of the four angles i.e., `angleA, angleB, angleC, and angleD` are `19=30^(@), 100^(@), 50^(@)`and `80^(@)`, respectively.