Question

given the image, which of these gives enough information to use one of the theorem converses to prove that p || r? m∠9 = 108°. m∠8 = 72°. m∠1 = 108°. m∠3 + m∠4 = 108°. two of these

Explanation:

Step1: Identify vertical - angles

∠9 and ∠7 are vertical - angles, so m∠7=m∠9 = 108°.

Step2: Use linear - pair relationship

∠7 and ∠8 form a linear pair. Since the sum of angles in a linear pair is 180°, we have m∠7 + m∠8=180°.

Step3: Solve for m∠8

Substitute m∠7 = 108° into m∠7 + m∠8=180°. Then 108°+m∠8 = 180°. So m∠8=180° - 108°=72°.
If ∠8 and a corresponding or alternate - interior angle on line p has the same measure, by the converse of corresponding - angles postulate or alternate - interior angles theorem, we can prove p∥r. So the information m∠8 = 72° is sufficient.

Answer:

m∠8 = 72°