Question

given the image, which of these gives enough information to use one of the theorem converses to prove that b || n? m∠7 = 68° m∠4 = 68° m∠5 = 112° m∠2 = 112° m∠6 = 112° 6 more questions to go

Explanation:

Step1: Recall parallel - line theorems

The converse of the alternate - interior angles theorem states that if two lines are cut by a transversal and the alternate - interior angles are congruent, then the two lines are parallel. The converse of the corresponding angles theorem states that if corresponding angles are congruent, then the lines are parallel. The converse of the same - side interior angles theorem states that if same - side interior angles are supplementary, then the lines are parallel.

Step2: Analyze angle relationships

We know that \(\angle7 = 68^{\circ}\). \(\angle2\) and \(\angle7\) are alternate - interior angles. If \(m\angle2=112^{\circ}\), then \(\angle2+\angle7 = 112^{\circ}+68^{\circ}=180^{\circ}\), which is not the condition for parallel lines based on alternate - interior angles. \(\angle4\) and \(\angle7\) are corresponding angles. If \(m\angle4 = 68^{\circ}\), then \(\angle4=\angle7\), and by the converse of the corresponding angles theorem, \(b\parallel n\). \(\angle5\) and \(\angle7\) are not related by the common parallel - line theorems in a way that directly proves \(b\parallel n\). \(\angle6\) and \(\angle7\) are same - side interior angles. If \(m\angle6 = 112^{\circ}\), then \(\angle6+\angle7=112^{\circ} + 68^{\circ}=180^{\circ}\), and by the converse of the same - side interior angles theorem, \(b\parallel n\).

Answer:

\(m\angle6 = 112^{\circ}\)