Question

in circle k, what is the value of x?
$\bigcirc$ $x = 30^\circ$
$\bigcirc$ $x = 25^\circ$
$\bigcirc$ $x = 20^\circ$
$\bigcirc$ $x = 15^\circ$

Explanation:

Step1: Identify triangle properties

The side through center $K$ is a diameter, so the triangle inscribed in the circle is a right triangle (Thales' theorem). Thus, one angle is $90^\circ$.

Step2: Sum angles to find $x$

The sum of angles in a triangle is $180^\circ$. Set up the equation:
$75^\circ + x^\circ + 90^\circ = 180^\circ$

Step3: Simplify and solve for $x$

Combine known angles: $165^\circ + x^\circ = 180^\circ$
Solve for $x$: $x = 180^\circ - 165^\circ = 15^\circ$

Answer:

$\boldsymbol{x = 15^\circ}$ (Option: $x = 15^\circ$)