Chapter 4.5, Problem 19PPS

To determine

To prove:

  $\angle KPL\cong \angle MRL$ for the following figure in the form of the two-column:

  

Explanation of Solution

Given information:

  $\begin{array}{l}\overline{KP}\perp \overline{PR}\\ \overline{MR}\perp \overline{PR}\\ \angle K\cong \angle M\end{array}$

Proof:

From the given figure:

  

The two-column proof is written as:

No.	Proof of steps	Reason
1.	$\begin{array}{l}\overline{KP}\perp \overline{PR}\\ \overline{MR}\perp \overline{PR}\\ \angle K\cong \angle M\end{array}$	Given information
2.	$\angle KPR=\angle MRP=90°$	According to the definition of perpendicularity
3.	$\angle KPR\cong \angle MRP$	Alternative right angles are always congruent in nature
4.	$\overline{PR}\cong \overline{PR}$	According to the reflexive property
5.	$\Delta KPR\cong \Delta MRP$	According to angle-angle-side (AAS) congruence postulate
6.	$\overline{KP}\cong \overline{MR}$	Corresponding parts of congruent triangles are congruent
7.	$\angle KLP\cong \angle MLR$	Vertical right angles are always congruent in nature
8.	$\Delta KLP\cong \Delta MLR$	According to angle-angle-side (AAS) congruence postulate
9.	$\angle KPL\cong \angle MRL$	Corresponding parts of congruent triangles are congruent

Hence proved