Question

given the image, which of these gives enough information to use one of the theorem converses to prove that b || n? m∠8 = 130°, m∠3 = 130°, two of these, m∠5 = 130°, m∠4 = 50°

Explanation:

Step1: Recall parallel - line theorems

If two lines are cut by a transversal, several angle - relationships can prove parallelism.

Step2: Analyze $m\angle3 = 130^{\circ}$

$\angle3$ and $\angle8$ are corresponding angles. Given $m\angle8 = 130^{\circ}$ and $m\angle3=130^{\circ}$, by the corresponding - angles converse (if corresponding angles are congruent, then the lines are parallel), $b\parallel n$.

Step3: Analyze $m\angle5 = 130^{\circ}$

$\angle5$ and $\angle8$ are alternate exterior angles. Since $m\angle8 = 130^{\circ}$ and $m\angle5 = 130^{\circ}$, by the alternate - exterior - angles converse (if alternate exterior angles are congruent, then the lines are parallel), $b\parallel n$.

Step4: Analyze $m\angle4 = 50^{\circ}$

$\angle4$ and $\angle8$ are same - side exterior angles. Given $m\angle8 = 130^{\circ}$ and $m\angle4 = 50^{\circ}$, since $m\angle4+m\angle8=50^{\circ}+130^{\circ}=180^{\circ}$, by the same - side exterior - angles converse (if same - side exterior angles are supplementary, then the lines are parallel), $b\parallel n$.

Answer:

All of the options ($m\angle3 = 130^{\circ}$, $m\angle5 = 130^{\circ}$, $m\angle4 = 50^{\circ}$) can be used to prove $b\parallel n$.