Question

in the diagram below, parallel lines l and k are cut by transversal p. the measure of angle 6 is 32°. what is m∠4 + m∠5? a. 296° b. 328° c. 64° d. 148°

Explanation:

Step1: Identify angle - relationships

Since lines \(l\) and \(k\) are parallel and \(p\) is a transversal, \(\angle6\) and \(\angle4\) are alternate - interior angles. So, \(m\angle4=m\angle6 = 32^{\circ}\).

Step2: Identify another angle - relationship

\(\angle5\) and \(\angle6\) are a linear pair. So, \(m\angle5 + m\angle6=180^{\circ}\), then \(m\angle5=180^{\circ}-m\angle6\). Substituting \(m\angle6 = 32^{\circ}\), we get \(m\angle5 = 180 - 32=148^{\circ}\).

Step3: Calculate \(m\angle4 + m\angle5\)

\(m\angle4 + m\angle5=32^{\circ}+148^{\circ}=180^{\circ}\). But there is a mistake above. Let's start over. \(\angle6\) and \(\angle2\) are corresponding angles, so \(m\angle2 = m\angle6=32^{\circ}\). And \(\angle2\) and \(\angle4\) are vertical angles, so \(m\angle4 = 32^{\circ}\). \(\angle5\) and \(\angle3\) are alternate - interior angles, and \(\angle3\) and \(\angle4\) are a linear pair. \(\angle5\) and \(\angle4\) are same - side interior angles. Since \(l\parallel k\), \(m\angle4 + m\angle5=180^{\circ}\).

Answer:

A. \(180^{\circ}\) (There seems to be an error in the provided options as the correct sum based on parallel - line properties is \(180^{\circ}\), but following the closest logic with the given options, we assume there was a mis - typing in the problem - setup and if we consider the correct angle - relationship calculations for the given framework, we note that the sum of same - side interior angles of parallel lines cut by a transversal is \(180^{\circ}\))