Question

1. numbered below are the 14 angles formed by parallel lines l and m and transversals p and q. lines l, p, and q intersect at a single point. which of the following congruence statements must be true? a. ∠1≅∠5 b. ∠2≅∠1 c. ∠3≅∠10 d. ∠4≅∠2 e. ∠5≅∠8

Explanation:

Step1: Recall angle - congruence rules for parallel lines and transversals

When two parallel lines are cut by transversals, vertical angles are congruent, corresponding angles are congruent, alternate - interior angles are congruent, and alternate - exterior angles are congruent.

Step2: Analyze option A

$\angle1$ and $\angle5$ are vertical angles. Vertical angles are always congruent. By the vertical - angle theorem, if two angles are vertical angles, then they have equal measures. So, $\angle1\cong\angle5$.

Step3: Analyze option B

$\angle2$ and $\angle1$ are adjacent angles. They are supplementary (sum to 180 degrees) but not congruent in general.

Step4: Analyze option C

$\angle3$ and $\angle10$ are neither vertical, corresponding, alternate - interior, nor alternate - exterior angles. There is no reason for them to be congruent based on the parallel - line and transversal properties.

Step5: Analyze option D

$\angle4$ and $\angle2$ are adjacent angles. They are supplementary (sum to 180 degrees) but not congruent in general.

Step6: Analyze option E

$\angle5$ and $\angle8$ are neither vertical, corresponding, alternate - interior, nor alternate - exterior angles. There is no reason for them to be congruent based on the parallel - line and transversal properties.

Answer:

A. $\angle1\cong\angle5$