Chapter 4.5, Problem 18WE

To determine

Show $\overline{AX}\cong \overline{BY}$

Explanation of Solution

Given:

  $\angle A\text{ and }\angle \text{B}$ are the base angles of isosceles $\Delta ABC$ , and the bisector of $\angle A$ meets $\overline{BC}$ at X and the bisector of $\angle B$ meets $\overline{AC}$ at Y.

Calculation:

Represent the given details :

  

Since triangle ABC is isosceles and $\angle A\cong \angle B$ ,so, $\overline{AC}\cong \overline{BC}$ .

Also , AX is angle bisector of $\angle A$ and BY is angle bisector of $\angle B$ .

And $\angle A=2\angle CAX\text{ and }\angle B=2\angle CBY$

  $\angle A\cong \angle B\Rightarrow \angle CAX\cong \angle CBY$

Consider $\Delta CAX\text{ and }\Delta CBY$ :

  $\angle C\cong \angle C$

  $\overline{CA}\cong \overline{CB}$

  $\angle CAX\cong \angle CBY$

So, by ASA Postulate , $\Delta CAX\cong \Delta CBY$

So, $\overline{AX}\cong \overline{BY}$

Hence proved.