In \( \triangle \mathrm{PQR}, \overline{P R} \) is extended through point R to point S , \( \mathrm{m} \angle P Q R=(3 x+13)^{\circ}, \mathrm{m} \angle Q R S=(7 x+7)^{\circ} \), and \( \mathrm{m} \angle R P Q=(x+12)^{\circ} \). Find \( \mathrm{m} \angle R P Q \).

To find \( \mathrm{m} \angle R P Q \), we start by using the fact that the angles around point R must sum up to \( 180^\circ \) since they are supplementary angles on a straight line. Therefore, we can set up the equation: \[ \mathrm{m} \angle P Q R + \mathrm{m} \angle Q R S = 180^\circ \] Substituting the values we have: \[ (3x + 13) + (7x + 7) = 180 \] Now, let's combine the terms: \[ 3x + 7x + 13 + 7 = 180 \] This simplifies to: \[ 10x + 20 = 180 \] Next, we can isolate \( x \): \[ 10x = 180 - 20 \] \[ 10x = 160 \] \[ x = 16 \] Now that we have \( x \), we can substitute it back into the expression for \( \mathrm{m} \angle R P Q \): \[ \mathrm{m} \angle R P Q = x + 12 \] \[ \mathrm{m} \angle R P Q = 16 + 12 = 28^\circ \] Thus, the measure of angle \( \mathrm{R P Q} \) is \[ \boxed{28^\circ}. \]