Question

which of the following is true to prove p||q ? a (mangle5cong mangle8) b (mangle1cong mangle6) c (mangle3 + mangle8=180^{circ}) d (mangle1 + mangle2 = 180^{circ})

Explanation:

Step1: Recall parallel - line postulates

If two lines are cut by a transversal, certain angle - relationships imply parallelism.

Step2: Analyze option A

$\angle5$ and $\angle8$ are vertical angles. Vertical - angle equality ($m\angle5\cong m\angle8$) does not prove $p\parallel q$.

Step3: Analyze option B

$\angle1$ and $\angle6$ are neither corresponding, alternate interior, nor alternate exterior angles. Their equality does not prove $p\parallel q$.

Step4: Analyze option C

$\angle3$ and $\angle8$ are same - side exterior angles. If the sum of same - side exterior angles is $180^{\circ}$, then the two lines ($p$ and $q$) cut by the transversal $m$ are parallel. That is, if $m\angle3 + m\angle8=180^{\circ}$, then $p\parallel q$.

Step5: Analyze option D

$\angle1$ and $\angle2$ are adjacent angles on a straight line. Their sum being $180^{\circ}$ ($m\angle1 + m\angle2 = 180^{\circ}$) is a property of linear pairs and does not prove $p\parallel q$.

Answer:

C. $m\angle3 + m\angle8 = 180^{\circ}$