## Elementary Geometry for College Students (6th Edition)

1. Prove that $\angle P\cong\angle R$ 2. Prove that $\overline{MP}\cong\overline{MR}$ 3. Prove that $\angle NMP\cong\angle QMR$ 4. Then $\triangle NMP\cong\triangle QMR$ according to method ASA 5. Finally $\angle N\cong\angle Q$ according to method CPCTC.
*PLANNING: To show that $\angle N\cong\angle Q$, we need to prove $\triangle NMP\cong\triangle QMR$. To prove that, we notice that - $\angle P$ and $\angle R$ are right angles, so they must be congruent. - M is the midpoint of $\overline{PR}$, so two lines created out of there must be congruent. - $\angle NMP\cong\angle QMR$ since they are 2 vertical angles. Therefore, we can use ASA to prove triangles congruent. 1. $\angle P$ and $\angle R$ are right angles. (Given) 2. $\angle P\cong\angle R$ (2 corresponding right angles are congruent) 3. M is the midpoint of $\overline{PR}$. (Given) 4. $\overline{MP}\cong\overline{MR}$ (A midpoint divides the line into 2 congruent lines) 5. $\angle NMP\cong\angle QMR$ (2 vertical angles are congruent with each other) So now we have 2 angles and the included side of $\triangle NMP$ are congruent with 2 corresponding angles and the included side of $\triangle QMR$. Therefore, 6. $\triangle NMP\cong\triangle QMR$ (ASA) 7. $\angle N\cong\angle Q$ (CPCTC)