Question

given the image, which of these gives enough information to use one of the theorem converses to prove that p || r? m∠2 = 74°. m∠6 = 63° and m∠10 = 43°. m∠6 = 29° and m∠10 = 45°. m∠6 = 17° and m∠10 = 57°. two of these

Explanation:

Step1: Recall parallel - line theorem

If two lines are cut by a transversal, and the alternate - interior angles are congruent, then the lines are parallel. $\angle6$ and $\angle10$ are alternate - interior angles.

Step2: Check angle congruence

For lines $p$ and $r$ to be parallel, we need $m\angle6=m\angle10$. In option B, $m\angle6 = 29^{\circ}$ and $m\angle10 = 45^{\circ}$, and $29^{\circ}
eq45^{\circ}$, so this option does not prove $p\parallel r$. In option A, $m\angle6 = 63^{\circ}$ and $m\angle10 = 43^{\circ}$, $63^{\circ}
eq43^{\circ}$, so it does not prove $p\parallel r$. In option C, $m\angle6 = 17^{\circ}$ and $m\angle10 = 57^{\circ}$, $17^{\circ}
eq57^{\circ}$, so it does not prove $p\parallel r$. Since no single option proves $p\parallel r$, the answer is "Two of these" as we need to check the angle - relationships more comprehensively with other angle pairs or additional information.

Answer:

B. $m\angle6 = 29^{\circ}$ and $m\angle10 = 45^{\circ}$