The complete two-column proof that proves that [tex]l \parallel m[/tex] is:
Statement 1: [tex]\angle 2 $ and $ \angle 5[/tex] are supplementary.
Reason: Given
Statement 2: [tex]\angle 2 $ \cong $ \angle 3[/tex]
Reason: Vertical angles.
Statement 3: [tex]\angle 2 $ and $ \angle 5[/tex] are supplementary
Reason: Substitution Property
Statement 4: [tex]l \parallel m[/tex]
Reason: Converse of Same-Side Interior Angles Theorem
The referred diagram has been attached below. See attachment.
Thus:
Statement 1: [tex]\angle 2 $ and $ \angle 5[/tex] are supplementary.
Reason: Given (we have been told already).
Statement 2: [tex]\angle 2 $ \cong $ \angle 3[/tex]
Reason: Vertical angles. (angle 2 and angle 3 are vertically opposite each other, and are therefore congruent to each other because they are vertical angles.)
Statement 3: [tex]\angle 2 $ and $ \angle 5[/tex] are supplementary
Reason: Substitution Property (Since we have proven that [tex]\angle 2 $ and $ \angle 5[/tex] are supplementary, and [tex]\angle 2 $ \cong $ \angle 3[/tex], then by the substitution property, [tex]\angle 3 $ and $ \angle 5[/tex] are supplementary.)
Statement 4: [tex]l \parallel m[/tex]
Reason: Converse of Same-Side Interior Angles Theorem (Since we have proven that [tex]\angle 3 $ and $ \angle 5[/tex] are supplementary, and both angles lie on same side along the transversal and are within the two lines intercepted by the transversal, therefore the Converse of Same-Side Interior Angle Theorem states that lines l and m will be parallel.).
In summary, the complete two-column proof that proves that [tex]l \parallel m[/tex] is:
Statement 1: [tex]\angle 2 $ and $ \angle 5[/tex] are supplementary.
Reason: Given
Statement 2: [tex]\angle 2 $ \cong $ \angle 3[/tex]
Reason: Vertical angles.
Statement 3: [tex]\angle 2 $ and $ \angle 5[/tex] are supplementary
Reason: Substitution Property
Statement 4: [tex]l \parallel m[/tex]
Reason: Converse of Same-Side Interior Angles Theorem
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