Question

find the measure of the missing angles.
answer attempt 1 out of 3
x = \boxed{\space}^\circ \quad y = \boxed{\space}^\circ

Explanation:

Step1: Find x (right angle)

The angle with \( 30^\circ \) and \( x \) forms a right angle (\( 90^\circ \)). So, \( x + 30^\circ = 90^\circ \). Solving for \( x \): \( x = 90^\circ - 30^\circ = 60^\circ \).

Step2: Find y (complementary to x)

The angle \( y \), \( x \), and the right angle (vertical) form a straight line? Wait, no—actually, \( y \) and \( x \) and the right angle? Wait, looking at the diagram, the horizontal and vertical lines are perpendicular, so the angle between horizontal left and vertical up is \( 90^\circ \), which is \( y + x = 90^\circ \). Wait, we found \( x = 60^\circ \), so \( y = 90^\circ - x = 90^\circ - 60^\circ = 30^\circ \)? Wait, no—wait, the vertical line (up) and the horizontal line (left-right) are perpendicular, so the angle between the left horizontal and the up vertical is \( 90^\circ \), which is split into \( y \) and \( x \). So \( y + x = 90^\circ \), so \( y = 90^\circ - x = 90 - 60 = 30^\circ \)? Wait, no, wait the \( 30^\circ \) is between the up vertical and the other line, and \( x \) is between that line and the horizontal? Wait, maybe better: the right angle (vertical and horizontal) is \( 90^\circ \), so the angle between the left horizontal and the up vertical is \( 90^\circ \), which is \( y + x = 90^\circ \). Since \( x = 60^\circ \) (from \( x + 30^\circ = 90^\circ \)), then \( y = 90^\circ - 60^\circ = 30^\circ \)? Wait, no, maybe I messed up. Wait, the vertical line (up) and the line with \( 30^\circ \) make \( 30^\circ \), so \( x \) is between that line and the horizontal (left-right). So the angle between up vertical and horizontal (right) is \( 90^\circ \), so \( x + 30^\circ = 90^\circ \), so \( x = 60^\circ \). Then, the angle between left horizontal and the line with \( x \) is \( y \), and since the left horizontal and up vertical is \( 90^\circ \), \( y + x = 90^\circ \), so \( y = 90 - 60 = 30^\circ \). Wait, but also, the \( 30^\circ \) and \( y \) should be equal? Wait, maybe the lines are symmetric? Wait, no, let's re-express. The horizontal line (left-right) and vertical line (up-down) are perpendicular, so the angle between left horizontal and up vertical is \( 90^\circ \). This angle is divided into \( y \) (between left horizontal and the slanted line) and \( x \) (between slanted line and up vertical). Then, there's another angle of \( 30^\circ \) between up vertical and another slanted line. Wait, maybe the two slanted lines are symmetric? No, the diagram shows one slanted line with \( 30^\circ \) from up vertical, and another slanted line with \( x \) from up vertical? Wait, no, the diagram: the up vertical line, then a slanted line with \( 30^\circ \) between them, then another slanted line with \( x \) between that and the horizontal (left) line? Wait, maybe the key is that the angle between the up vertical and the horizontal (right) is \( 90^\circ \), so \( x + 30^\circ = 90^\circ \), so \( x = 60^\circ \). Then, the angle \( y \) is between the left horizontal and the slanted line (with \( x \)), so since the left horizontal and up vertical is \( 90^\circ \), \( y + x = 90^\circ \), so \( y = 90^\circ - 60^\circ = 30^\circ \). Alternatively, maybe \( y \) is equal to \( 30^\circ \) because of vertical angles or something, but the calculation via \( x + 30 = 90 \) (so \( x = 60 \)) and \( y + x = 90 \) (so \( y = 30 \)) makes sense.

Answer:

\( x = 60^\circ \), \( y = 30^\circ \)