Question

1. which lines can you conclude are parallel given that m∠7 + m∠11 = 180? justify your conclusion with a theorem.

line a is parallel to line b by the converse of the same - side interior angles theorem.
line c is parallel to line d by the converse of the same - side interior angles theorem.
line a is parallel to line b by the converse of the alternate interior angles theorem.
line c is parallel to line d by the converse of the alternate interior angles theorem.

Explanation:

Step1: Identify angle - pair type

$\angle7$ and $\angle11$ are same - side interior angles formed by lines $a$, $b$ and transversal $d$.

Step2: Apply the converse of the theorem

The Converse of the Same - Side Interior Angles Theorem states that if two lines are cut by a transversal and the same - side interior angles are supplementary (sum to 180°), then the two lines are parallel. Since $m\angle7 + m\angle11=180^{\circ}$, lines $a$ and $b$ are parallel.

Answer:

Line a is parallel to line b by the Converse of the Same - Side Interior Angles Theorem.