Question

the diagram shows two parallel lines cut by a transversal. one angle measure is shown. explain how you determined the measure of angle z.

Explanation:

Step1: Identify Angle Relationship

When two parallel lines are cut by a transversal, vertical angles are equal, and consecutive interior angles (or supplementary angles) sum to \(180^\circ\). Also, corresponding angles are equal. First, find the angle adjacent to the given angle (if it's a linear pair) or use vertical/corresponding angles.

Step2: Use Supplementary Angles (if needed)

Suppose the given angle (let's say \(x = 30^\circ\) from the diagram's hint) and angle \(z\) form a linear pair (supplementary) or are corresponding. If the given angle and angle \(z\) are supplementary (sum to \(180^\circ\)), then \(z = 180^\circ - 30^\circ = 150^\circ\). Or if vertical angles, they are equal. But typically, with parallel lines and transversal, if one angle is \(30^\circ\), its supplementary angle (linear pair) would be \(150^\circ\), and angle \(z\) (if corresponding or vertical to that supplementary angle) would be \(150^\circ\) or equal to the given if vertical. Wait, let's clarify: If the given angle is \(30^\circ\), the angle vertical to it is \(30^\circ\), but angle \(z\) – if the lines are parallel, corresponding angles: if the given angle's supplementary angle (linear pair, \(180 - 30 = 150\)) is a corresponding angle to \(z\), then \(z = 150^\circ\). So the key is: 1. Find the angle adjacent (linear pair) to the given angle: \(180 - \text{given angle}\). 2. Since lines are parallel, corresponding angles are equal, so angle \(z\) equals that supplementary angle. So if given angle is \(30^\circ\), linear pair angle is \(150^\circ\), and \(z\) (corresponding) is \(150^\circ\).

Answer:

To determine angle \( z \): 1. The given angle (e.g., \( 30^\circ \)) and its linear - pair angle are supplementary (\( 180^\circ - 30^\circ=150^\circ \)). 2. Since the lines are parallel, \( z \) and this supplementary angle are corresponding angles (or vertical angles, depending on diagram), so \( z = 150^\circ \) (assuming the given angle is \( 30^\circ \); adjust based on actual diagram - shown angle). The process uses properties of parallel lines (corresponding angles, supplementary angles) and linear pairs.