Question

what is the value of x?

Explanation:

Step1: Recognize isosceles triangle

Since $HI = HJ$ (the two - line segments are marked as equal), $\triangle HIJ$ is isosceles. In an isosceles triangle, the base - angles are equal. Let's assume the measure of $\angle I=\angle J = 15x$.

Step2: Recall angle - sum property of a triangle

The sum of the interior angles of a triangle is $180^{\circ}$. In $\triangle HIJ$, if we let the third - angle be $\angle H$. Since $IJ$ is a diameter of the circle, the angle subtended by a diameter at the circumference is a right - angle, so $\angle H = 90^{\circ}$.

Step3: Set up an equation

Using the angle - sum property of a triangle ($\angle I+\angle J+\angle H=180^{\circ}$), we substitute the values: $15x + 15x+90^{\circ}=180^{\circ}$.

Step4: Simplify the equation

Combine like terms: $30x+90 = 180$. Subtract 90 from both sides: $30x=180 - 90$, so $30x = 90$.

Step5: Solve for x

Divide both sides of the equation $30x = 90$ by 30: $x=\frac{90}{30}=3$.

Answer:

$3$