Question

find the measure of the missing angles.
answer attempt 1 out of 2
x = \square^\circ \quad y = \square^\circ

Explanation:

Step1: Find x (right angle)

A right angle is \( 90^\circ \), so \( x + 41^\circ = 90^\circ \). Solve for \( x \): \( x = 90 - 41 = 49^\circ \).

Step2: Find y (vertical angles/right angle)

The angle with \( y \) and \( x + 41^\circ \) is also a right angle? Wait, no—horizontal and vertical lines are perpendicular, so the angle between horizontal and vertical is \( 90^\circ \). Wait, actually, \( y + x + 41^\circ = 90^\circ \)? No, wait, the vertical line and horizontal line form a right angle. Wait, looking at the diagram, the horizontal line and vertical line are perpendicular (right angle). So the angle between the left horizontal and the vertical line's left side: \( y + x = 90^\circ \)? Wait, no, earlier we found \( x = 49^\circ \), and since \( y \) and the angle with \( x \) and \( 41^\circ \)—wait, actually, \( y \) is equal to \( 41^\circ \)? No, wait, no. Wait, the vertical line (up and down) and horizontal line (left and right) are perpendicular, so the angle between them is \( 90^\circ \). The angle between the upper vertical (up) and the right arrow is \( 41^\circ \), and \( x \) is between that and the left upper arrow. Then, the left horizontal and the left upper arrow: \( y \) is equal to \( 41^\circ \)? Wait, no, let's re-express.

Wait, the right angle (between horizontal and vertical) is \( 90^\circ \). So the angle composed of \( x \), \( 41^\circ \), and the right angle? No, the vertical line (up) and the right horizontal form a right angle? Wait, no, the vertical line (up and down) and horizontal line (left and right) are perpendicular, so the angle between the up vertical and right horizontal is \( 90^\circ \). So \( x + 41^\circ = 90^\circ \), so \( x = 49^\circ \). Then, the angle \( y \) is equal to \( 41^\circ \)? Wait, no, because the left horizontal and the left upper arrow: since the lines are symmetric? Wait, no, actually, the angle between the left horizontal and the left upper arrow ( \( y \)) should be equal to \( 41^\circ \)? Wait, no, that's not right. Wait, no—actually, the angle \( y \) and \( x \) and the right angle? Wait, no, the horizontal line (left and right) and vertical line (up and down) are perpendicular, so the angle between left horizontal and up vertical is \( 90^\circ \). So \( y + x = 90^\circ \). Since \( x = 49^\circ \), then \( y = 90 - 49 = 41^\circ \)? Wait, no, that contradicts. Wait, no, let's start over.

Wait, the key is that the vertical line (up) and the right horizontal line form a right angle ( \( 90^\circ \) ). So the angle between the up vertical and the right arrow is \( 41^\circ \), so the remaining angle ( \( x \)) between the up vertical and the left arrow (upper left) is \( 90 - 41 = 49^\circ \). Then, the angle between the left horizontal line and the upper left arrow ( \( y \)) is equal to the angle between the right horizontal line and the upper right arrow ( \( 41^\circ \) ), because they are vertical angles or alternate interior angles? Wait, no, actually, the horizontal line is straight, and the vertical line is straight, so the angle \( y \) is equal to \( 41^\circ \)? No, that can't be. Wait, no, the angle \( y \) is equal to \( 41^\circ \)? Wait, no, let's think again.

Wait, the horizontal line (left and right) and vertical line (up and down) are perpendicular, so the angle between left horizontal and up vertical is \( 90^\circ \). The angle between up vertical and upper right arrow is \( 41^\circ \), so the angle between upper right arrow and right horizontal is \( 90 - 41 = 49^\circ \)? No, that's not. Wait, I'm getting confused. Let's use the fact that the sum of angles around a point is \( 360^\circ \), but here we have right angles. Wait, the vertical line (up and down) and horizontal line (left and right) intersect at a right angle, so the angle between them is \( 90^\circ \). So the angle composed of \( x \), \( 41^\circ \), and the right angle? No, the up vertical and right horizontal: the angle between them is \( 90^\circ \), so \( x + 41^\circ = 90^\circ \), so \( x = 49^\circ \). Then, the angle \( y \) is equal to \( 41^\circ \), because the lines are symmetric? Wait, no, actually, \( y \) is equal to \( 41^\circ \)? Wait, no, that's not. Wait, the angle \( y \) and \( x \) and the right angle: \( y + x = 90^\circ \), so \( y = 90 - 49 = 41^\circ \). Ah, there we go. So \( y = 41^\circ \), \( x = 49^\circ \).

Wait, let's verify: \( x = 49^\circ \), \( 41^\circ + 49^\circ = 90^\circ \) (right angle), correct. Then \( y + x = 90^\circ \), so \( y = 90 - 49 = 41^\circ \), correct. So \( x = 49^\circ \), \( y = 41^\circ \).

Answer:

\( x = \boxed{49}^\circ \), \( y = \boxed{41}^\circ \)