Question

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

Explanation:

Step1: Identify Vertical Angles

The given angle (let's say it's \( x = 38^\circ \)) and the angle adjacent to angle \( C \) (or related by parallel lines and transversal) – first, vertical angles are equal. So the angle vertical to \( 38^\circ \) is also \( 38^\circ \).

Step2: Use Corresponding Angles (Parallel Lines)

Since the lines are parallel and cut by a transversal, corresponding angles are equal. The angle equal to the vertical angle of \( 38^\circ \) (from step 1) will be equal to angle \( C \) if they are corresponding angles. Alternatively, if the given angle and angle \( C \) are alternate - interior or alternate - exterior angles, they are equal. Also, if the given angle and angle \( C \) are supplementary (linear pair), but in the case of parallel lines cut by transversal, if the given angle is \( 38^\circ \), and angle \( C \) is equal (corresponding/alternate) or supplementary. Wait, let's correct: If the given angle is \( 38^\circ \), and we assume the angle adjacent to it (linear pair) is \( 180 - 38=142^\circ \), but no – let's think again. If two parallel lines are cut by a transversal, corresponding angles are equal. So if the given angle is \( 38^\circ \), and angle \( C \) is a corresponding angle, then angle \( C = 38^\circ \). Or if the given angle and angle \( C \) are vertical angles - no, vertical angles are formed by intersecting lines. Wait, the diagram has two parallel lines cut by a transversal, so the angle with measure \( 38^\circ \) and angle \( C \): if the angle with \( 38^\circ \) and angle \( C \) are corresponding angles (because of parallel lines), then angle \( C = 38^\circ \). Or if the given angle and angle \( C \) are alternate - interior angles, they are equal. So first, find the angle that is equal to the given \( 38^\circ \) (vertical angles are equal, so the angle opposite to \( 38^\circ \) is \( 38^\circ \)), then since the lines are parallel, corresponding angles are equal, so angle \( C \) is equal to that \( 38^\circ \) angle.

Answer:

To determine the measure of angle \( C \):

1. First, identify the angle vertical to the given \( 38^\circ \) angle. Vertical angles are equal, so this vertical angle is also \( 38^\circ \).
2. Since the two lines are parallel and cut by a transversal, corresponding angles (or alternate - interior/alternate - exterior angles) are equal. The angle equal to the \( 38^\circ \) vertical angle (from step 1) and angle \( C \) are corresponding (or alternate) angles. Thus, angle \( C = 38^\circ \) (assuming the given angle is \( 38^\circ \) and angle \( C \) is a corresponding/alternate angle to the vertical angle of \( 38^\circ \)). If the given angle and angle \( C \) are supplementary (linear pair), then angle \( C=180 - 38 = 142^\circ \), but based on parallel lines and transversal, more likely corresponding angles, so angle \( C = 38^\circ \). (The exact measure depends on the diagram's configuration, but the process is: use vertical angles and parallel - line angle properties (corresponding/alternate/supplementary) to find angle \( C \).)