Answer
(a) $m\angle P = 59^{\circ}$
(b) $NR = 4~in$
Work Step by Step
(a) Since $\overline{MN} \parallel \overline{QP}$, the measure of the angle $m\angle M = 90^{\circ}$
We can find $m\angle P$:
$m\angle P + m\angle Q + m\angle M + m\angle MNP = 360^{\circ}$
$m\angle P + m\angle Q + m\angle M + (m\angle M+31^{\circ}) = 360^{\circ}$
$m\angle P + 90^{\circ} + 90^{\circ}+ (90^{\circ}+31^{\circ}) = 360^{\circ}$
$m\angle P + 301^{\circ}) = 360^{\circ}$
$m\angle P = 59^{\circ}$
(b) We can find $PR$:
$PR = PQ- QR$
$PR = PQ-MN$
$PR = 9~in-6~in$
$PR = 3~in$
We can find the length of $\overline{NR}$:
$NR = \sqrt{(NP)^2-(PR)^2}$
$NR = \sqrt{(5~in)^2-(3~in)^2}$
$NR = \sqrt{25~in^2-9~in^2}$
$NR = \sqrt{16~in^2}$
$NR = 4~in$
