We are given a cyclic quadrilateral $PQRS$ inscribed in a circle. The angles are given as follows: $\angle P = x$, $\angle Q = 2y - 30^\circ$, $\angle R = x + y$, $\angle S = x$. We need to find the value of $x$.

1. Problem Description

We are given a cyclic quadrilateral $PQRS$ inscribed in a circle. The angles are given as follows: $\angle P = x$, $\angle Q = 2y - 30^\circ$, $\angle R = x + y$, $\angle S = x$. We need to find the value of $x$.

2. Solution Steps

In a cyclic quadrilateral, opposite angles are supplementary, meaning they add up to $180^\circ$. Therefore, we have two equations:

$$\angle P + \angle R = 180^\circ$$

$$\angle Q + \angle S = 180^\circ$$

Substituting the given values, we get:

$$x + (x + y) = 180$$

$$2x + y = 180$$

$$2y - 30 + x = 180$$

$$x + 2y = 210$$

Now we have a system of two linear equations with two variables:

$$2x + y = 180$$

$$x + 2y = 210$$

We can multiply the first equation by 2 to eliminate $y$:

$$4x + 2y = 360$$

$$x + 2y = 210$$

Subtract the second equation from the modified first equation:

$$(4x + 2y) - (x + 2y) = 360 - 210$$

$$3x = 150$$

$$x = \frac{150}{3}$$

$$x = 50$$

Now we can find the value of $y$:

$$2(50) + y = 180$$

$$100 + y = 180$$

$$y = 80$$

We can verify our solution by plugging the values of $x$ and $y$ into the second equation:

$$50 + 2(80) = 50 + 160 = 210$$

So, the value of $x$ is $50^\circ$.

3. Final Answer

The value of $x$ is $50^\circ$.

A. $50^\circ$