Chapter 4.6, Problem 9WE

To determine

Prove $\overline{SK}\cong \overline{YL}$

Explanation of Solution

Given:

  $\Delta RST\cong \Delta XYZ$

  $\overrightarrow{SK}\text{ bisects }\angle \text{RST}$

  $\overrightarrow{YL}\text{ bisects }\angle XYZ$

Calculation:

The diagram is :

  

Since $\Delta RST\cong \Delta XYZ$ ,

  $\angle RST\cong \angle XYZ$

  $\overline{ST}\cong \overline{YZ}$

  $\angle T\cong \angle Z$

Now,

  $\overrightarrow{SK}\text{ bisects }\angle \text{RST}$ and $\overrightarrow{YL}\text{ bisects }\angle XYZ$

So, $\angle \text{RST}=2\angle TSK$ and $\angle XYZ=2\angle ZYL$

Since $\angle RST\cong \angle XYZ\Rightarrow \angle TSK\cong \angle ZYL$

Consider the $\Delta SKT\text{ and }\Delta YLZ$ :

  $\angle T\cong \angle Z$

  $\overline{ST}\cong \overline{YZ}$

  $\angle TSK\cong \angle ZYL$

So, by ASA Postulate , $\Delta SKT\cong \Delta YLZ$

Hence , $\overline{SK}\cong \overline{YL}$

Hence proved.