Question

1. there are two parallel lines cut by a transversal. one angle is labeled ( x + 74 ) and another is labeled ( 112^circ ).

Explanation:

Step1: Identify Angle Relationship

The two angles \( x + 74 \) and \( 112^\circ \) are same - side interior angles (or consecutive interior angles) formed by a transversal intersecting two parallel lines. For parallel lines, same - side interior angles are supplementary, meaning their sum is \( 180^\circ \). So we can set up the equation \( (x + 74)+112 = 180 \).

Step2: Solve for \( x \)

First, simplify the left - hand side of the equation: \( x+74 + 112=x + 186 \).
Now we have the equation \( x + 186=180 \).
Subtract 186 from both sides of the equation: \( x=180 - 186=- 6 \)? Wait, that can't be right. Wait, maybe the angles are alternate interior angles or corresponding angles. Wait, maybe I misidentified the angle relationship. Let's re - examine the diagram. If the two lines are parallel, and the angle \( x + 74 \) and \( 112^\circ \) are actually equal (corresponding angles or alternate interior angles). Let's assume that the two lines are parallel and the transversal cuts them, so \( x + 74=112 \).
Now, solve for \( x \): Subtract 74 from both sides. \( x=112 - 74 = 38 \). Wait, maybe the first assumption of supplementary was wrong. Let's check the diagram again. If the two angles are on the same side of the transversal and the lines are parallel, but maybe the angle \( x + 74 \) and \( 112^\circ \) are equal (corresponding angles). Let's redo the step.
If the two angles are equal (corresponding angles, since the lines are parallel), then \( x + 74=112 \).
Subtract 74 from both sides: \( x=112 - 74=38 \).

Answer:

\( x = 38 \)