The angles of a cyclic quadrilateral ABCD are A=(6x+10), B=(5x), C=(x+y), D=(3y-10). Then, which of the following is the values of x, y and the values of the four angles?

Consider the given question, the angles of a cyclic quadrilateral

ABCD

are

A = (6 x + 10)

B = (5 x)

C = (x + y)

D = (3 y - 10)

.
Construct a figure using the given data,

In a cyclic quadrilateral sum of opposite angles are supplementary.
Then,

∠ A + ∠ C = 180^{o}

⟹ (6 x + 10 °) + (x + y) = 180 °

⟹ 6 x + 10 ° + x + y = 180 °

⟹ 7 x + 10 ° + y = 180 °

⟹ 7 x + y = 180 ° - 10 °

⟹ 7 x + y = 170 °

……..(1)
Similarly,

∠ B + ∠ D = 180^{o}

⟹

(3 y - 10 °) + 5 x = 180^{o}

⟹

3 y + 5 x = 180^{o} + 10 °

⟹

3 y + 5 x = 190^{o}

……..(2)
Solving (1) and (2), we have,

x = 20^{o}

y = 30^{o}

Then,

∠ A = 6 x + 10

\Rightarrow 6 \times 20^{o} + 10^{o}

\Rightarrow ∠ A = 130^{o}

∠ B = 5 x

\Rightarrow 5 \times 20^{o}

\Rightarrow ∠ B = 100^{o}

∠ C = x + y

\Rightarrow 30^{o} + 20^{o}

\Rightarrow ∠ C =

50^{o}

∠ D = 3 y - 10

\Rightarrow 3 \times 30^{o} - 10^{o}

\Rightarrow ∠ D = 80^{o}

Hence, the value of

x = 20 °, y = 30 °, A = 130^{o}, B = 100^{o}, C = 50^{o}

and

D = 80^{o}

.
Therefore, option 2 is correct.

The angles of a cyclic quadrilateral $ABCD$ are $A = (6 x + 10)$ , $B = (5 x)$ , $C = (x + y)$ , $D = (3 y - 10)$ . Then, which of the following is the values of $x$ , $y$ and the values of the four angles?