Question

which statement must always be true? choose the correct answer below. a. if ∠2≅∠4, then j∥k. b. if ∠1≅∠2, then g∥h. c. if ∠1≅∠3, then g∥h. d. if ∠3≅∠4, then

Explanation:

Step1: Recall parallel - line postulates

If two lines are cut by a transversal, alternate - interior angles being congruent implies parallel lines. $\angle2$ and $\angle4$ are alternate - interior angles for lines $j$ and $k$ with transversal $h$.

Step2: Analyze each option

• Option A: $\angle2$ and $\angle4$ are alternate - interior angles for lines $j$ and $k$. If $\angle2\cong\angle4$, then by the Alternate - Interior Angles Theorem, $j\parallel k$.
• Option B: $\angle1$ and $\angle2$ are not angles that can be used to prove $g\parallel h$ directly. They are not corresponding, alternate - interior, or alternate - exterior angles for lines $g$ and $h$.
• Option C: $\angle1$ and $\angle3$ are vertical angles, and their congruence does not imply $g\parallel h$.
• Option D: $\angle3$ and $\angle4$ are not angles that can be used to prove $g\parallel h$ directly. They are not corresponding, alternate - interior, or alternate - exterior angles for lines $g$ and $h$.

Answer:

A. If $\angle2\cong\angle4$, then $j\parallel k$.