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? o ∠1≅∠2 o ∠2≅∠3 o ∠1 and ∠2 are supplementary o ∠1 and ∠3 are supplementary

Explanation:

Step1: Recall parallel - line postulates

When two lines are cut by a transversal, if corresponding angles are congruent, alternate - interior angles are congruent, or same - side interior angles are supplementary, the two lines are parallel.

Step2: Analyze each option

• Option 1: $\angle1\cong\angle2$ is not a valid condition for parallel lines. $\angle1$ and $\angle2$ are not corresponding, alternate - interior, or same - side interior angles.
• Option 2: $\angle2\cong\angle3$ is not a valid condition for parallel lines. $\angle2$ and $\angle3$ are not corresponding, alternate - interior, or same - side interior angles.
• Option 3: $\angle1$ and $\angle2$ being supplementary is not a valid condition for parallel lines. $\angle1$ and $\angle2$ are not same - side interior angles.
• Option 4: If $\angle1$ and $\angle3$ are supplementary, and $\angle1$ and $\angle3$ are same - side interior angles formed by lines $a$ and $b$ and transversal $l$, then by the same - side interior angles postulate, lines $a$ and $b$ are parallel.

Answer:

$\angle1$ and $\angle3$ are supplementary