Question

find the measure of the missing angles.

Explanation:

Step1: Find angle \( x \)

The angle between the vertical line and the horizontal line is \( 90^\circ \) (right angle). So \( x + 53^\circ = 90^\circ \). Solving for \( x \), we get \( x = 90^\circ - 53^\circ = 37^\circ \).

Step2: Find angle \( y \)

Angle \( y \) and the angle \( x \) (which we found as \( 37^\circ \)) and the right angle (vertical and horizontal) - wait, actually, angle \( y \) and the angle between the left - slanting line and the vertical line (which is \( x = 37^\circ \)) and the right angle? Wait, no. The horizontal and vertical lines are perpendicular, so the angle between the left - slanting line and the horizontal line (\( y \)) and the angle between the left - slanting line and the vertical line (\( x = 37^\circ \)) should add up to \( 90^\circ \). Wait, actually, since the vertical and horizontal lines are perpendicular (\( 90^\circ \)), and the angle between the vertical line and the right - slanting line is \( 53^\circ \), the angle between the vertical line and the left - slanting line is \( x = 37^\circ \) (from step 1). Then, the angle between the left - slanting line and the horizontal line (\( y \)) and the angle between the left - slanting line and the vertical line (\( x \)) should add up to \( 90^\circ \) (because horizontal and vertical are perpendicular). So \( y + x=90^\circ \), and since \( x = 37^\circ \), then \( y=90^\circ - 37^\circ = 53^\circ \)? Wait, no, wait. Wait, the vertical line is perpendicular to the horizontal line, so the angle between vertical and horizontal is \( 90^\circ \). The angle between vertical and the right - slanting line is \( 53^\circ \), so the angle between vertical and left - slanting line is \( x = 90^\circ - 53^\circ=37^\circ \). Then, the angle between left - slanting line and horizontal line (\( y \)) and the angle between left - slanting line and vertical line (\( x \)) sum to \( 90^\circ \) (because horizontal and vertical are perpendicular). So \( y + x=90^\circ \), so \( y = 90^\circ - x=90 - 37 = 53^\circ \)? Wait, no, maybe a better way: the angle between the left - slanting line and the horizontal line (\( y \)) and the angle between the left - slanting line and the vertical line (\( x \)) and the right angle? Wait, no, the horizontal and vertical lines are perpendicular, so the angle between them is \( 90^\circ \). The left - slanting line divides this \( 90^\circ \) angle into \( y \) and \( x \). So \( y + x=90^\circ \). We found \( x = 37^\circ \), so \( y = 90 - 37=53^\circ \)? Wait, or maybe the angle \( y \) is equal to the angle of \( 53^\circ \) because of vertical angles? Wait, no, let's re - examine.

Wait, the right - slanting line makes a \( 53^\circ \) angle with the vertical line. The left - slanting line: since the vertical line is perpendicular to the horizontal line, the angle between the left - slanting line and the vertical line is \( x \), and the angle between the left - slanting line and the horizontal line is \( y \). Since vertical and horizontal are perpendicular (\( 90^\circ \)), \( x + y=90^\circ \). Also, the angle between the vertical line and the right - slanting line is \( 53^\circ \), and since the two slanting lines are symmetric? Wait, no, maybe the angle \( y \) is equal to \( 53^\circ \) and \( x = 37^\circ \). Wait, let's do step by step:

1. For angle \( x \): The angle between the vertical line and the right - slanting line is \( 53^\circ \), and the angle between the vertical line and the left - slanting line is \( x \). Since the vertical line is perpendicular to the horizontal line, the sum of the angle between vertical and right - slanting (\( 53^\circ \)) and the angle between vertical and left - slanting (\( x \)) should be \( 90^\circ \) (because the angle between vertical and horizontal is \( 90^\circ \), and the two slanting lines are on either side of the vertical line within that \( 90^\circ \) angle? Wait, no, actually, the vertical line and the horizontal line form a right angle (\( 90^\circ \)). The right - slanting line is inside this right angle, making \( 53^\circ \) with the vertical line. The left - slanting line is also inside this right angle, making \( x \) with the vertical line. So \( x+53^\circ = 90^\circ \), so \( x = 90 - 53=37^\circ \).

2. For angle \( y \): The angle between the horizontal line and the left - slanting line is \( y \), and the angle between the horizontal line and the vertical line is \( 90^\circ \). The angle between the vertical line and the left - slanting line is \( x = 37^\circ \), so \( y + x=90^\circ \), so \( y=90 - 37 = 53^\circ \). Alternatively, since the angle between the horizontal line and the right - slanting line would be \( 53^\circ \) (because the angle between vertical and right - slanting is \( 53^\circ \), and vertical and horizontal are \( 90^\circ \), so \( 90 - 53 = 37^\circ \)? Wait, no, I think I messed up. Wait, let's use the fact that the vertical line is perpendicular to the horizontal line, so the angle between them is \( 90^\circ \). The right - slanting line: angle with vertical is \( 53^\circ \), so angle with horizontal is \( 90 - 53 = 37^\circ \). The left - slanting line: angle with vertical is \( x \), angle with horizontal is \( y \). Since the two slanting lines: the angle between the left - slanting and right - slanting lines? Wait, no, the diagram shows a vertical line, a horizontal line, and two slanting lines (left and right) from the intersection point. The intersection point has a right angle between vertical and horizontal. The right - slanting line makes \( 53^\circ \) with vertical, so the angle between right - slanting and horizontal is \( 90 - 53 = 37^\circ \). The left - slanting line: since the angle between vertical and left - slanting is \( x \), and the angle between vertical and horizontal is \( 90^\circ \), then \( x + y=90^\circ \) (where \( y \) is the angle between left - slanting and horizontal). Also, since the two slanting lines: the angle between left - slanting and right - slanting should be \( x + 53^\circ \). But maybe a simpler way: the angle \( y \) is equal to \( 53^\circ \) and \( x = 37^\circ \), because the angle between the left - slanting line and the horizontal line is equal to the angle between the right - slanting line and the vertical line (alternate angles or something), but actually, from the right angle (vertical and horizontal), \( x+53 = 90 \), so \( x = 37 \), and \( y + x=90 \), so \( y = 53 \).

Answer:

\( x = 37^\circ \), \( y = 53^\circ \)