Question

given the image, is there enough information to prove that p || r? m∠3 = 42°, m∠4 = 68°, m∠7 = 110°. options: yes, by the alternate exterior angles theorem converse; no, this is not enough information; yes, by the alternate interior angles theorem converse; yes, by the consecutive interior angles theorem converse.

Explanation:

Step1: Analyze angle - relationships

We are given \(m\angle3 = 42^{\circ}\), \(m\angle4=68^{\circ}\), \(m\angle7 = 110^{\circ}\). To prove \(p\parallel r\) using angle - based parallel line theorems (alternate exterior, alternate interior, consecutive interior), we need relevant angle pairs.

Step2: Check alternate exterior angles

There is no information about a pair of alternate exterior angles formed by \(p\) and \(r\) that would allow us to use the alternate exterior angles theorem converse.

Step3: Check alternate interior angles

There is no information about a pair of alternate interior angles formed by \(p\) and \(r\) that would allow us to use the alternate interior angles theorem converse.

Step4: Check consecutive interior angles

There is no information about a pair of consecutive interior angles formed by \(p\) and \(r\) that would allow us to use the consecutive interior angles theorem converse. So, there is not enough information to prove \(p\parallel r\).

Answer:

No, this is not enough information.