Chapter 5.1, Problem 43PPS

To determine

To prove: The relation $\overline{PD}\cong \overline{PE}$ .

Explanation of Solution

Given information:

The given triangle shown in figure (1).

  

Figure (1)

Th given relations are.

  $\overline{AP}$ is a bisects of $\angle BAC$ ; $\overline{PD}\perp \overline{AB}$ and $\overline{PE}\perp \overline{AC}$ .

Proof:

Using two column proof.

Statement	Reason
$\overline{AP}$ is a bisects of   $\angle BAC$  $\overline{PD}\perp \overline{AB}$$\overline{PE}\perp \overline{AC}$	Given
$\angle DAP\cong \angle EAP$	Definition of angle bisector
$\angle PDA$$\angle PEA$ are right angles	Definition of perpendicular lines
$\angle PDA\cong \angle PEA$	Right angles are congruent
$\overline{AP}\cong \overline{AP}$	Reflexive property of congruence
$\Delta EAP\cong \Delta DAP$	AAS Congruence theorem
$\overline{PD}\cong \overline{PE}$	CPCTC

From the above, $\overline{PD}\cong \overline{PE}$ .

Hence, proved.