Question

find the measure of \\( \angle 4 \\), and state the postulate or theorem used to justify your answer.

please select the best answer from the choices provided

Explanation:

Step1: Recall the Consecutive Interior Angles Theorem

The Consecutive Interior Angles Theorem states that if two parallel lines are cut by a transversal, then the sum of the measures of consecutive interior angles is \(180^{\circ}\).
In the given figure, lines \(m\) and \(n\) are parallel (as indicated by the arrow markings), and line \(l\) is a transversal.
Let the angle adjacent to the \(65^{\circ}\) angle be \(\angle x\). We know that \(\angle x + 65^{\circ}=180^{\circ}\) (linear - pair of angles). But we can also use the Consecutive Interior Angles Theorem.
If we consider the two parallel lines \(m\) and \(n\) cut by transversal \(l\), the \(65^{\circ}\) angle and \(\angle4\) are consecutive interior angles.

Step2: Calculate the measure of \(\angle4\)

By the Consecutive Interior Angles Theorem, \(m\angle4+65^{\circ}=180^{\circ}\).
Subtract \(65^{\circ}\) from both sides of the equation: \(m\angle4 = 180^{\circ}-65^{\circ}\).

$$m\angle4=115^{\circ}$$

Answer:

C. \(115^{\circ}\), Consecutive Interior Angles Theorem.