Chapter 5.6, Problem 29E

To determine

To Show: The $\Delta MNP\sim \Delta RST$ .

Explanation of Solution

Given:For $\Delta MNPand\Delta RST,\angle M\cong \angle R,\angle N\cong \angle S,\angle P\cong \angle T.$

Given: $\Delta MNPand\Delta RST,\angle M\cong \angle R,\angle N\cong \angle S,\angle P\cong \angle T.$

Then

Using the law of sines.

  $\begin{array}{l}\frac{m}{\sin M}=\frac{n}{\mathrm{sinN}}\\ \sin M=\frac{m\sin N}{n}\\ \frac{r}{\mathrm{sinR}}=\frac{s}{\mathrm{sinS}}\\ \sin R=\frac{r\sin S}{s}\\ \sin ce\\ \angle M\cong \angle R\\ thus\\ \frac{m\sin N}{n}=\frac{r\mathrm{sinS}}{s}\\ \sin ce\\ \angle N\cong \angle S\\ \sin N=\sin S\\ so\\ \frac{m}{n}=\frac{r}{s}\\ \frac{m}{r}=\frac{n}{s}\end{array}$

Similar proportions can also be derived for $pandt$ .

Thus

  $\Delta MNP\sim \Delta RST$

Hence proved.