Question

the measure of $angle kjn$ is $92^\circ$. what is the measure of $overparen{jmk}$?

Explanation:

Step1: Recall tangent-chord angle theorem

The measure of an angle formed by a tangent and a chord is half the measure of its intercepted arc. Here, $\angle KJN$ intercepts arc $\overparen{JK}$.

Step2: Calculate measure of $\overparen{JK}$

If $m\angle KJN = \frac{1}{2}m\overparen{JK}$, then $m\overparen{JK} = 2 \times m\angle KJN = 2 \times 92^\circ = 184^\circ$.

Step3: Find measure of $\overparen{JMK}$

A full circle is $360^\circ$, so subtract $\overparen{JK}$ from $360^\circ$:
$m\overparen{JMK} = 360^\circ - m\overparen{JK}$
$m\overparen{JMK} = 360^\circ - 184^\circ$

Answer:

$176^\circ$