1. **State the problem:** Given that triangles $\triangle POR$ and $\triangle STU$ are similar, find the missing side lengths $ST$, $TU$ and the angle measures $m\angle S$, $m\angle T$, $m\angle U$ in $\triangle STU$. 2. **Recall properties of similar triangles:** Corresponding angles are equal and corresponding sides are proportional. 3. **Identify corresponding parts:** Since $\triangle POR \sim \triangle STU$, we have: - $P \leftrightarrow S$ - $O \leftrightarrow T$ - $R \leftrightarrow U$ Therefore: - $m\angle P = m\angle S$ - $m\angle O = m\angle T$ - $m\angle R = m\angle U$ And sides correspond as: - $PO \leftrightarrow ST$ - $OR \leftrightarrow TU$ - $PR \leftrightarrow SU$ 4. **Given values in $\triangle POR$:** - $m\angle P = 70^\circ$ - $m\angle R = 46^\circ$ - $PO = 14$ - $OR = 28$ - $PR = 21$ 5. **Find $m\angle O$ in $\triangle POR$:** $$m\angle O = 180^\circ - m\angle P - m\angle R = 180^\circ - 70^\circ - 46^\circ = 64^\circ$$ 6. **Find missing angles in $\triangle STU$ using angle correspondence:** - $m\angle S = m\angle P = 70^\circ$ - $m\angle U = m\angle R = 46^\circ$ - $m\angle T = m\angle O = 64^\circ$ 7. **Use side length correspondence and scale factor:** Given $SU = 6$ corresponds to $PR = 21$, the scale factor $k$ from $\triangle POR$ to $\triangle STU$ is: $$k = \frac{SU}{PR} = \frac{6}{21} = \frac{2}{7}$$ 8. **Find $ST$ and $TU$ using scale factor:** - $ST$ corresponds to $PO = 14$, so: $$ST = k \times PO = \frac{2}{7} \times 14 = 4$$ - $TU$ corresponds to $OR = 28$, so: $$TU = k \times OR = \frac{2}{7} \times 28 = 8$$ **Final answers:** - $ST = 4$ - $TU = 8$ - $m\angle S = 70^\circ$ - $m\angle T = 64^\circ$ - $m\angle U = 46^\circ$