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 parallel, certain angle relationships hold. For lines \(j\) and \(k\) cut by transversals \(m\) and \(n\), we use the properties of corresponding, alternate - interior, alternate - exterior, and same - side interior angles.

Step2: Analyze each option

• Option A: \(\angle2=\angle3\) are vertical angles. Vertical angles are equal regardless of whether \(j\parallel k\) or not.
• Option B: \(\angle1 = \angle4\) are vertical angles. Vertical angles are equal regardless of whether \(j\parallel k\) or not.
• Option C: \(\angle2\) and \(\angle5\) are same - side interior angles. According to the same - side interior angles postulate, if two lines are cut by a transversal, the two same - side interior angles are supplementary (\(m\angle2 + m\angle5=180^{\circ}\)) if and only if the two lines are parallel.
• Option D: \(\angle6=\angle4\) are vertical angles. Vertical angles are equal regardless of whether \(j\parallel k\) or not.

Answer:

C. \(m\angle2 + m\angle5 = 180\)