We are given a circle with triangle $YWX$ inscribed in it. We are given that $\angle YWX = (x+14)^\circ$ and $\angle YXW = x^\circ$. We want to find the value of $x$.

1. Problem Description

We are given a circle with triangle $YWX$ inscribed in it. We are given that $\angle YWX = (x+14)^\circ$ and $\angle YXW = x^\circ$. We want to find the value of $x$.

2. Solution Steps

Since $WX$ is a diameter of the circle, the angle $\angle WYX$ is a right angle.

Therefore, $\angle WYX = 90^\circ$.

The sum of the angles in any triangle is $180^\circ$.

Therefore, in $\triangle YWX$, we have

$$\angle YWX + \angle YXW + \angle WYX = 180^\circ$$

Substituting the given values, we have

$$(x+14) + x + 90 = 180$$

$$2x + 104 = 180$$

$$2x = 180 - 104$$

$$2x = 76$$

$$x = \frac{76}{2}$$

$$x = 38$$

3. Final Answer

$x = 38$