Question

what is the value of x? x - 8°

Explanation:

Step1: Identify triangle type

Since the two sides of the triangle inscribed in the circle are equal (marked with the same cross - hatch), it is an isosceles triangle. So, $\angle K=\angle J=x - 8^{\circ}$.

Step2: Apply angle - sum property of a triangle

The sum of the interior angles of a triangle is $180^{\circ}$. Let the third angle be $\angle I$. For an isosceles triangle $\triangle IKJ$, we have $\angle I + \angle K+\angle J=180^{\circ}$. Since the base angles $\angle K=\angle J=x - 8^{\circ}$, and assuming the central angle subtended by the arc $IJ$ and $IK$ is such that the triangle is formed within the circle, we know that $\angle I + 2(x - 8^{\circ})=180^{\circ}$. Also, if we consider the fact that the triangle is inscribed in a semi - circle (diameter is $KJ$), $\angle I = 90^{\circ}$ (angle inscribed in a semi - circle is a right angle).
Substitute $\angle I = 90^{\circ}$ into the equation: $90+(x - 8)+(x - 8)=180$.

Step3: Simplify the equation

$90 + 2x-16 = 180$.
$2x+74 = 180$.
$2x=180 - 74$.
$2x = 106$.

Step4: Solve for x

$x=\frac{106}{2}=53$.

Answer:

$53$