Question

1. which statements are enough to conclude that m || n? a. ∠6≅∠3 b. ∠1 and ∠7 are supplementary. c. ∠1≅∠9 d. ∠1≅∠12

Explanation:

Step1: Recall parallel - line postulates

If two lines are cut by a transversal, corresponding angles are equal, alternate - interior angles are equal, or same - side interior angles are supplementary, then the two lines are parallel.

Step2: Analyze option a

$\angle6$ and $\angle3$ are alternate - interior angles. If $\angle6\cong\angle3$, by the Alternate Interior Angles Theorem, $m\parallel n$.

Step3: Analyze option b

$\angle1$ and $\angle7$ are not related in a way that can directly prove $m\parallel n$. $\angle1$ and $\angle7$ are not corresponding, alternate - interior, or same - side interior angles with respect to lines $m$ and $n$ and transversals $p$ or $q$.

Step4: Analyze option c

$\angle1$ and $\angle9$ are corresponding angles with respect to lines $m$ and $n$ and transversals $p$ and $q$. If $\angle1\cong\angle9$, by the Corresponding Angles Postulate, $m\parallel n$.

Step5: Analyze option d

$\angle1$ and $\angle12$ are not related in a way that can directly prove $m\parallel n$. $\angle1$ and $\angle12$ are not corresponding, alternate - interior, or same - side interior angles with respect to lines $m$ and $n$ and transversals $p$ or $q$.

Answer:

A. $\angle6\cong\angle3$
C. $\angle1\cong\angle9$