Chapter 4.5, Problem 2WE

To determine

To prove: $\overline{LK}\cong \overline{JK}$

Explanation of Solution

Given information: $\overline{KL}\perp \overline{LA},\overline{KJ}\perp \overline{JA}$ and $\overrightarrow{AK}$ bisects $\angle LAJ$ .

Proof:

  

Step (a)

  $\overline{KL}\perp \overline{LA},\overline{KJ}\perp \overline{JA}$ (Given)

Step (d)

  $m\angle L=90°,m\angle J=90°$ (two lines are said to be perpendicular to each other if they form an angle of $90°$ )

Step (f)

  $\angle L\cong \angle J$ (Right angles are congruent to each other)

Step (c)

  $\overrightarrow{AK}$ bisects $\angle LAJ$ (Given)

Step (b)

  $\angle 1\cong \angle 2$ (bisector divides an angle into congruent angles)

Step (h)

  $\overline{KA}\cong \overline{KA}$ (reflexive property of congruence)

(according to reflexive property of congruence, an angle is congruent to itself)

Step (g)

  $\Delta LKA\cong \Delta JKA$ (AAS congruence criteria)

(According to AAS congruence criteria, two angles are said to be congruent if two angles and the side not included between them of one triangle are congruent to the two angles and the side not included between them of another triangle)

Step (e)

  $\overline{LK}\cong \overline{JK}$ (Corresponding parts of congruent triangles are equal)