Question

1. given: < 1 ≅< 4 and < 3 & < 4 are supplementary prove: l||m statements reasons 1. < 1 ≅< 4 and < 3 & < 4 are supp 1. given 2. < 2 ≅ < 3 2. vertical 3. < 1 and < 2 are supp 3. alt int to < 3 and < 4 are sub 4. 4. 5. 5. 6. 6.

Explanation:

Step1: Recall vertical - angle property

$\angle1\cong\angle4$ (given), $\angle2\cong\angle3$ (vertical angles are congruent)

Step2: Use substitution and supplementary - angle property

Since $\angle3$ and $\angle4$ are supplementary ($\angle3+\angle4 = 180^{\circ}$) and $\angle1\cong\angle4$, $\angle2\cong\angle3$, then $\angle1$ and $\angle2$ are supplementary ($\angle1+\angle2=180^{\circ}$)

Step3: Apply same - side interior angles postulate

If two lines are cut by a transversal and a pair of same - side interior angles are supplementary, then the two lines are parallel. Since $\angle1$ and $\angle2$ are same - side interior angles for lines $l$ and $m$ cut by transversal $r$ and $\angle1+\angle2 = 180^{\circ}$, we have $l\parallel m$

Answer:

1. $l\parallel m$; Same - side interior angles are supplementary
2. N/A (not needed as the proof is complete at step 4)
3. N/A (not needed as the proof is complete at step 4)