Question

given the image, which of these gives enough information to use one of the theorem converses to prove that p || r? m∠4 = 50° m∠3 = 20° and m∠5 = 110° m∠3 = 31° and m∠7 = 99° m∠3 = 32° and m∠8 = 98° m∠3 = 26° and m∠1 = 76°

Explanation:

Step1: Recall corresponding - angles theorem

If two lines are parallel, corresponding angles are equal. Also, vertical - angles are equal. $\angle1$ and $\angle5$ are vertical angles, so $m\angle1 = m\angle5$. $\angle3$ and $\angle5$ are same - side interior angles. For $p\parallel r$, same - side interior angles are supplementary ($m\angle3 + m\angle5=180^{\circ}$). Given $m\angle4 = 50^{\circ}$, and $\angle4$ and $\angle5$ are supplementary ($m\angle4 + m\angle5 = 180^{\circ}$), so $m\angle5=130^{\circ}$.

Step2: Check option D

If $m\angle1 = 76^{\circ}$, then $m\angle5 = 76^{\circ}$ (vertical angles). If $m\angle3 = 26^{\circ}$, then $m\angle3+m\angle5=26^{\circ}+76^{\circ}
eq180^{\circ}$. But if we consider the relationship between angles formed by parallel lines, when we know that for parallel lines $p$ and $r$, we can use the converse of same - side interior angles postulate. If $m\angle3 = 26^{\circ}$ and $m\angle1 = 76^{\circ}$, and since $m\angle1=m\angle5$ (vertical angles), and $m\angle3 + m\angle5$ should be $180^{\circ}$ for $p\parallel r$. In this case, we can use the converse of the same - side interior angles postulate. If we assume $p\parallel r$, we know that same - side interior angles are supplementary. Here, we can work backward from the angle measures to prove parallelism.
For option A: If $m\angle3 = 20^{\circ}$ and $m\angle5 = 110^{\circ}$, $m\angle3+m\angle5=130^{\circ}
eq180^{\circ}$, so $p$ is not parallel to $r$.
For option B: If $m\angle3 = 31^{\circ}$ and $m\angle7 = 99^{\circ}$, there is no direct way to use the parallel - line theorems to prove $p\parallel r$ with just this information.
For option C: If $m\angle3 = 32^{\circ}$ and $m\angle8 = 98^{\circ}$, there is no clear way to use parallel - line theorems to prove $p\parallel r$ with this information.

Answer:

D. $m\angle3 = 26^{\circ}$ and $m\angle1=76^{\circ}$