Question

in circle p with $m\angle nrq = 18^\circ$, find the $m\angle npq$.

Explanation:

Step1: Identify arc for inscribed angle

$\angle NRQ$ is an inscribed angle intercepting arc $NQ$. The measure of an inscribed angle is half the measure of its intercepted arc.
$m\overset{\frown}{NQ} = 2 \times m\angle NRQ$

Step2: Calculate intercepted arc measure

Substitute $m\angle NRQ = 18^\circ$
$m\overset{\frown}{NQ} = 2 \times 18^\circ = 36^\circ$

Step3: Relate central angle to arc

$\angle NPQ$ is a central angle intercepting arc $NQ$. The measure of a central angle equals the measure of its intercepted arc.
$m\angle NPQ = m\overset{\frown}{NQ}$

Answer:

$36^\circ$