Question

which statement must be true to prove j || k?
a. ∠2 ≅ ∠3
b. ∠1 ≅ ∠4
c. m∠2 + m∠5 = 180
d. ∠6 ≅ ∠4

Explanation:

Step1: Recall parallel - line postulates

When two lines are cut by a transversal, certain angle - relationships imply parallel lines.

Step2: Analyze option A

$\angle2$ and $\angle3$ are vertical angles. Vertical angles are always congruent, but their congruence does not prove that $j\parallel k$.

Step3: Analyze option B

$\angle1$ and $\angle4$ are vertical angles. Their congruence does not prove that $j\parallel k$.

Step4: Analyze option C

$\angle2$ and $\angle5$ are same - side interior angles. According to the same - side interior angles postulate, if the sum of two same - side interior angles is $180^{\circ}$, then the two lines are parallel. But here, $m\angle2 + m\angle5=180$ is incorrect for proving $j\parallel k$ as they are not the correct pair of same - side interior angles for lines $j$ and $k$.

Step5: Analyze option D

$\angle6$ and $\angle4$ are alternate interior angles. According to the alternate interior angles theorem, if two alternate interior angles are congruent when two lines are cut by a transversal, then the two lines are parallel.

Answer:

D. $\angle6\cong\angle4$