Question

given the image, which of these gives enough information to use one of the theorem converses to prove that p || r? m∠8 = 65°. m∠7 = 115°. two of these. m∠3 + m∠4 = 65°. m∠3 + m∠4 = 115°.

Explanation:

Step1: Recall vertical - angle property

$\angle8$ and $\angle6$ are vertical angles, so $m\angle6=m\angle8 = 65^{\circ}$.

Step2: Recall linear - pair property

$\angle6$ and $\angle7$ form a linear pair. Since the sum of angles in a linear pair is $180^{\circ}$, we have $m\angle6 + m\angle7=180^{\circ}$.

Step3: Substitute and solve

Substitute $m\angle6 = 65^{\circ}$ into $m\angle6 + m\angle7=180^{\circ}$. Then $65^{\circ}+m\angle7 = 180^{\circ}$, so $m\angle7=180^{\circ}- 65^{\circ}=115^{\circ}$.
If $m\angle7 = 115^{\circ}$, and $\angle7$ and $\angle3$ are corresponding angles, when corresponding angles are equal, the lines $p$ and $r$ are parallel (corresponding - angles converse).

Answer:

$m\angle7 = 115^{\circ}$