Question

given the following information, determine which lines, if any, are parallel. state the theorem that justifies your answer.

$m\angle5 + m\angle6 = 180^\circ$

\\(\bigcirc\\) a) $a \parallel b$; consecutive interior angles converse
\\(\bigcirc\\) b) $c \parallel d$; consecutive exterior angles converse
\\(\bigcirc\\) c) $a \parallel b$; alternate interior angles converse
\\(\bigcirc\\) d) $c \parallel d$; alternate exterior angles converse

Explanation:

Step1: Identify Angles and Lines

∠5 and ∠6 are consecutive interior angles formed by transversal (the middle line) intersecting lines \(a\) and \(b\).

Step2: Apply Theorem

Given \(m\angle5 + m\angle6 = 180^\circ\), by Consecutive Interior Angles Converse (if consecutive interior angles are supplementary, lines are parallel), \(a \parallel b\).

Answer:

A) \(a \parallel b\); Consecutive Interior Angles Converse