Question

in the figure, lines a and b are intersected by line t. which of these statements proves that lines a and b are parallel? ∠2≅∠3 ∠1 and ∠3 are supplementary ∠1 and ∠2 are supplementary ∠1≅∠2

Explanation:

Step1: Recall parallel - line postulates

When two lines are cut by a transversal, if alternate - interior angles are congruent, the lines are parallel.

Step2: Identify angle types

In the given figure, $\angle2$ and $\angle3$ are alternate - interior angles. If $\angle2\cong\angle3$, then by the Alternate - Interior Angles Theorem, lines $a$ and $b$ are parallel.

Step3: Analyze other options

• $\angle1$ and $\angle3$ being supplementary does not prove $a\parallel b$.
• $\angle1$ and $\angle2$ being supplementary is a property of adjacent angles on a straight line and does not prove $a\parallel b$.
• $\angle1\cong\angle2$ is a property of vertical angles and does not prove $a\parallel b$.

Answer:

$\angle2\cong\angle3$