Elementary Geometry for College Students (6th Edition)

1. Prove that $\angle N\cong\angle Q$ and $\angle P\cong\angle R$ 2. Prove that $\overline{MN}\cong\overline{MQ}$ 3. Then $\triangle NMP\cong\triangle QMR$ according to method AAS 4. Then $\overline{NP}\cong\overline{QR}$ according to CPCTC
*PLANNING: To show that $\overline{NP}\cong\overline{QR}$, we need to prove $\triangle NMP\cong\triangle QMR$. To prove that, we notice that - $\overline{NP}\parallel\overline{RQ}$ with transversals $\overline{PR}$ and $\overline{NQ}$, so 2 congruent pairs of alternate interior angles could be found. - M is the midpoint of $\overline{NQ}$, so two lines created out of there must be congruent. After that, we would have 2 congruent pairs of angles and 1 congruent pair of side. So we could use ASA or AAS to prove triangles congruent. 1. $\overline{NP}\parallel\overline{RQ}$ with transversals $\overline{PR}$ and $\overline{NQ}$. (Given) 2. $\angle N\cong\angle Q$ and $\angle P\cong\angle R$ (2 alternate interior angles are congruent) 3. M is the midpoint of $\overline{NQ}$. (Given) 4. $\overline{MN}\cong\overline{MQ}$ (A midpoint divides the line into 2 congruent lines) So now we have 2 angles and a non-included side of $\triangle NMP$ are congruent with 2 corresponding angles and a non-included side of $\triangle QMR$. Therefore, 6. $\triangle NMP\cong\triangle QMR$ (AAS) 7. $\overline{NP}\cong\overline{QR}$ (CPCTC)