1. **Problem 23:** Given $MN \cong QO$, $MN \parallel QO$, $NR \perp MP$, and $OP \perp MP$, prove $NR \cong OP$. 2. Since $NR \perp MP$ and $OP \perp MP$, by definition of perpendicular lines, both $\angle NR$ and $\angle OP$ are right angles. 3. All right angles are congruent, so $\angle NR \cong \angle OP$. 4. Given $MN \cong QO$ and $MN \parallel QO$, and $NR$ and $OP$ are perpendicular to $MP$, triangles $NRM$ and $OPQ$ are right triangles sharing a common side $MP$. 5. By the Hypotenuse-Leg (HL) theorem for right triangles, since $MN \cong QO$ (hypotenuses) and $\angle NR \cong \angle OP$ (right angles), and $MP$ is common, triangles $NRM$ and $OPQ$ are congruent. 6. Therefore, corresponding legs $NR$ and $OP$ are congruent: $NR \cong OP$. 7. **Problem 24:** Given $ST \cong WT$, $WTQ \cong STV$, $ST \cong VT$, and $QT \cong WT$, prove $RT \cong UT$. 8. Since $WTQ \cong STV$, corresponding parts of congruent triangles are congruent (CPCTC). 9. Given $ST \cong WT$ and $QT \cong WT$, triangles $RTQ$ and $UTV$ share congruent sides. 10. By Side-Angle-Side (SAS) or other triangle congruence criteria, triangles $RTQ$ and $UTV$ are congruent. 11. Therefore, corresponding sides $RT$ and $UT$ are congruent: $RT \cong UT$.