Question

given: \\( \angle 3 \\) and \\( \angle 5 \\) are supplementary.
prove: \\( l \parallel m \\)

statements

1. \\( \angle 3 \\) and \\( \angle 5 \\) are supplementary
2. \\( \angle 5 \\) and \\( \angle 6 \\) are supplementary

3.
4.

a. \\( l \parallel t \\)
b. \\( \angle 3 \cong \angle 6 \\)
c. def. of supplementary angles
d. \\( l \parallel m \\)
e. linear pairs theorem

reasons

1. given

2.

1. congruent supplements theorem
2. converse of the alternate interior angles theorem

Explanation:

Step1: Identify the reason for statement 2

$\angle5$ and $\angle6$ form a linear pair. By the Linear Pairs Theorem, they are supplementary.

Step2: Use the Congruent Supplements Theorem

Since $\angle3$ and $\angle5$ are supplementary ($\angle3+\angle5 = 180^{\circ}$) and $\angle5$ and $\angle6$ are supplementary ($\angle5+\angle6=180^{\circ}$), then $\angle3\cong\angle6$ (Congruent Supplements Theorem: If two angles are supplementary to the same angle, then they are congruent).

Step3: Apply the Converse of the Alternate Interior Angles Theorem

If $\angle3\cong\angle6$ (alternate - interior angles), then by the Converse of the Alternate Interior Angles Theorem, $l\parallel m$.

Answer:

1. e. Linear Pairs Theorem; 3. b. $\angle3\cong\angle6$; 4. d. $l\parallel m$