Question

1. diagram with two horizontal lines and a vertical transversal, angles (x + 28)° and (y - 18)°

x = ______
reason: __________________________
y = ______
reason: __________________________

1. diagram with two parallel horizontal lines and a transversal, angles (7y - 62)° and (5x)°

Explanation:

For \( x \):

Step1: Identify angle relationship

The angle \((x + 25)^\circ\) and the right angle ( \(90^\circ\)) are supplementary? No, wait, the first line and the vertical line form a right angle, so \((x + 25)^\circ\) and \(90^\circ\) are complementary? Wait, no, the first horizontal line and the vertical line are perpendicular, so the angle \((x + 25)^\circ\) and the right angle ( \(90^\circ\))? Wait, no, the angle \((x + 25)^\circ\) is adjacent to the right angle, so they should add up to \(180^\circ\)? No, wait, horizontal line and vertical line are perpendicular, so the angle between them is \(90^\circ\). Wait, the first angle is \((x + 25)^\circ\) and it's adjacent to the right angle, so actually, \((x + 25)^\circ + 90^\circ = 180^\circ\)? No, that would be if they are supplementary. Wait, no, the horizontal line is straight, so the sum of \((x + 25)^\circ\) and the right angle ( \(90^\circ\)) should be \(180^\circ\) because they are on a straight line. So:
\[
x + 25 + 90 = 180
\]

Step2: Solve for \( x \)

Simplify the left side: \(x + 115 = 180\)
Subtract 115 from both sides: \(x = 180 - 115 = 65\)

Step1: Identify angle relationship

The two horizontal lines are parallel (since they are both horizontal), and the vertical line is a transversal. The angle \((y - 18)^\circ\) and the right angle ( \(90^\circ\))? Wait, no, the two horizontal lines are parallel, so the corresponding angles should be equal. Wait, the first horizontal line and the vertical line form a right angle, so the second horizontal line and the vertical line should also form a right angle (since the lines are parallel and the transversal is vertical). Wait, no, the angle \((y - 18)^\circ\) is adjacent to the vertical line, so it should be equal to \(90^\circ\)? Wait, no, let's think again. The two horizontal lines are parallel, and the vertical line is perpendicular to the first horizontal line, so it should also be perpendicular to the second horizontal line. Therefore, the angle \((y - 18)^\circ\) should be equal to \(90^\circ\)? Wait, no, maybe the angle \((y - 18)^\circ\) and the right angle are equal because of parallel lines and transversal (perpendicular transversal implies corresponding angles are right angles). Wait, let's set up the equation:
\[
y - 18 = 90
\]

Step2: Solve for \( y \)

Add 18 to both sides: \(y = 90 + 18 = 108\)? Wait, no, that can't be. Wait, maybe the two horizontal lines are parallel, so the angle \((x + 25)^\circ\) and \((y - 18)^\circ\) are equal? Wait, no, the first angle is \((x + 25)^\circ = 65 + 25 = 90^\circ\), so \((y - 18)^\circ\) should also be \(90^\circ\) because the lines are parallel and the transversal is vertical, so corresponding angles are equal. So:
\[
y - 18 = 90
\]
Then \(y = 90 + 18 = 108\)? Wait, but let's check. If \(y = 108\), then \(y - 18 = 90\), which matches the right angle. So that makes sense.

Answer:

\( x = 65 \)
Reason: The angle \((x + 25)^\circ\) and the right angle (\(90^\circ\)) are supplementary (they form a linear pair), so their sum is \(180^\circ\). Solving \(x + 25 + 90 = 180\) gives \(x = 65\).

For \( y \):