3. If \( S \) is the incenter of \( \triangle P Q R, m \angle P R Q=8 x-10, m \angle R P Q=4 x-14 \), and \( m \angle P Q R=7 x-5 \), find each missing measure. \( \begin{array}{ll}P & \text { a) } m \angle P R Q=\end{array} \)

To solve for the missing angle measures, we first use the fact that the sum of the angles in a triangle is always \(180^\circ\). So, we can set up the equation: \[ m \angle P R Q + m \angle R P Q + m \angle P Q R = 180^\circ. \] Substituting the expressions in terms of \(x\): \[ (8x - 10) + (4x - 14) + (7x - 5) = 180. \] Combining like terms: \[ 8x + 4x + 7x - 10 - 14 - 5 = 180, \] \[ 19x - 29 = 180. \] Adding \(29\) to both sides gives: \[ 19x = 209. \] Dividing by \(19\): \[ x = 11. \] Now we can find each angle: 1. For \( m \angle P R Q \): \[ m \angle P R Q = 8x - 10 = 8(11) - 10 = 88 - 10 = 78^\circ. \] 2. For \( m \angle R P Q \): \[ m \angle R P Q = 4x - 14 = 4(11) - 14 = 44 - 14 = 30^\circ. \] 3. For \( m \angle P Q R \): \[ m \angle P Q R = 7x - 5 = 7(11) - 5 = 77 - 5 = 72^\circ. \] So, we have: - \( m \angle P R Q = 78^\circ \) - \( m \angle R P Q = 30^\circ \) - \( m \angle P Q R = 72^\circ \) Thus, the answers for the missing measures are: a) \( m \angle P R Q = 78^\circ \)