Question

what is the value of x? x = \boxed{\circ}

Explanation:

Step1: Identify triangle properties

$\triangle VUT$ is isosceles ($VU=TU$), so $\angle TVU = \angle VTU = x+30^\circ$. Also, $VT$ is a diameter, so $\angle VUT = 90^\circ$ (Thales' theorem).

Step2: Sum triangle angles

The sum of angles in a triangle is $180^\circ$.
$\angle TVU + \angle VTU + \angle VUT = 180^\circ$

Step3: Substitute values and solve

Substitute the known angles:
$$(x+30^\circ) + (x+30^\circ) + 90^\circ = 180^\circ$$
Simplify:
$$2x + 150^\circ = 180^\circ$$
$$2x = 180^\circ - 150^\circ$$
$$2x = 30^\circ$$
$$x = \frac{30^\circ}{2} = 15^\circ$$

Answer:

$15$