1. **Problem Statement:** We are given triangle $PQR$ with altitudes $\overline{RS}$ and $\overline{PT}$ drawn from vertices $R$ and $P$ respectively. The measure of angle $Q$ is $47^\circ$. We need to find the measure of angle $POR$. 2. **Key Concepts:** - Altitudes in a triangle are perpendicular to the opposite sides. - Since $\overline{RS}$ and $\overline{PT}$ are altitudes, $RS \perp PQ$ and $PT \perp QR$. - The point $O$ is the intersection of the altitudes, so $O$ is the orthocenter of triangle $PQR$. 3. **Step-by-step Solution:** - Since $\overline{PT}$ is an altitude from $P$ to $QR$, $PT \perp QR$. - Since $\overline{RS}$ is an altitude from $R$ to $PQ$, $RS \perp PQ$. - The orthocenter $O$ lies at the intersection of these altitudes. 4. **Using angle properties:** - In triangle $PQR$, the sum of interior angles is $180^\circ$. - Given $m\angle Q = 47^\circ$, so $m\angle P + m\angle R = 180^\circ - 47^\circ = 133^\circ$. 5. **Relationship of angles at orthocenter:** - The angle $POR$ is formed by the lines from $P$ and $R$ to the orthocenter $O$. - It is a known property that the angle between the altitudes at vertices $P$ and $R$ equals $180^\circ - m\angle Q$. 6. **Calculate $m\angle POR$:** $$ m\angle POR = 180^\circ - m\angle Q = 180^\circ - 47^\circ = 133^\circ $$ **Final answer:** $$ m\angle POR = 133^\circ $$