8. Write a two-column proof.
Given: $\overline{M P}$ is perpendicular to $\overline{Q R}$. $N$ is the midpoint of $\overline{M P} \cdot \overline{Q P} \cong \overline{R M}$ Prove: $\triangle M N R \cong \triangle P N Q$

Step 1 ：Given: $\overline{M P}$ is perpendicular to $\overline{Q R}$. $N$ is the midpoint of $\overline{M P} \cdot \overline{Q P} \cong \overline{R M}$

Step 2 ：This means that angle MPR and angle QPR are right angles.

Step 3 ：Also, since N is the midpoint of MP, MN = NP.

Step 4 ：Additionally, QP is congruent to RM, which means that QP = RM.

Step 5 ：We are asked to prove that triangle MNR is congruent to triangle PNQ.

Step 6 ：We can use the Hypotenuse-Leg (HL) Congruence Theorem, which states that if the hypotenuse and one leg of a right triangle are congruent to the hypotenuse and one leg of another right triangle, then the triangles are congruent.

Step 7 ：In this case, MN = NP (given), RM = QP (given), and angle MNR and angle PNQ are right angles (since MP is perpendicular to QR).

Step 8 ：Therefore, by the HL Congruence Theorem, triangle MNR is congruent to triangle PNQ.

Step 9 ：Final Answer: \(\boxed{\triangle M N R \cong \triangle P N Q}\)