Question

complete the two - column proof. complete parts a. through d.
given: \\( \angle s \cong \angle k, \overline{k j} \\| \overline{w s} \\)
prove: \\( \triangle s j w \cong \triangle k w j \\)

statements	reasons
2) \\( \overline{k j} \\	\overline{w s} \\)	2) a.
3) \\( \angle s w j \cong \angle k j w \\)	3) b.
4) c.	4) reflexive property
5) \\( \triangle s j w \cong \triangle k w j \\)	5) d.

Explanation:

Step1: Fill in reason for \( \overline{KJ}\parallel\overline{WS} \)

Since \( \overline{KJ}\parallel\overline{WS} \) is given in the problem statement.

Step2: Fill in reason for \( \angle SWJ\cong\angle KJW \)

When two parallel lines (\( \overline{KJ}\) and \( \overline{WS} \)) are cut by a transversal (\( \overline{JW} \)), alternate - interior angles are congruent.

Step3: Fill in statement for reflexive property

The reflexive property of congruence states that a segment is congruent to itself. So, \( \overline{JW}\cong\overline{JW} \).

Step4: Fill in reason for \( \triangle SJW\cong\triangle KWJ \)

We have two angles (\( \angle S\cong\angle K\), \( \angle SWJ\cong\angle KJW \)) and the included side (\( \overline{JW}\cong\overline{JW} \)) congruent. By the Angle - Angle - Side (AAS) congruence criterion.

Answer:

a. Given
b. Alternate - interior angles theorem (If two parallel lines are cut by a transversal, then alternate - interior angles are congruent)
c. \( \overline{JW}\cong\overline{JW} \)
d. AAS (Angle - Angle - Side) congruence criterion