Answer
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.
Work Step by Step
*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)
