Question

find the measure of the missing angles.

Explanation:

Step1: Determine angle \( x \)

We can see from the diagram that there is a right angle (90°) formed by the vertical and horizontal lines. The angle \( x \) and the 45° angle are complementary to form this right angle. So we use the property that the sum of complementary angles is 90°.
\( x + 45^\circ = 90^\circ \)
\( x = 90^\circ - 45^\circ = 45^\circ \)

Step2: Determine angle \( y \)

We can observe that angle \( y \), angle \( x \) (which we found to be 45°), and the right angle (90°) are related such that they form a straight line (180°). Wait, actually, looking at the horizontal and the lines, angle \( y \) and angle \( x \) and the right angle? Wait, no, actually, the horizontal line is a straight line (180°), and we have a right angle (90°) on one side, and angle \( y \), angle \( x \) on the other. Wait, more simply, angle \( y \) and angle \( x \) are complementary to the right angle? Wait, no, looking at the diagram, the angle between the horizontal line (left - right) and the line making angle \( y \) and \( x \) with the vertical. Wait, actually, since \( x = 45^\circ \), and the angle between the horizontal line (left) and the line with angle \( y \) and \( x \) is such that angle \( y \) and angle \( x \) are equal? Wait, no, let's re - examine. The vertical line (up - down) and horizontal line (left - right) are perpendicular (90°). The angle between the upper vertical line and the line with 45° is 45°, and angle \( x \) is also 45° (as we found). Then, the angle \( y \) is equal to angle \( x \) because of the symmetry? Wait, no, actually, the horizontal line (left - right) and the line making angle \( y \) with the horizontal and angle \( x \) with the vertical. Since the vertical and horizontal are perpendicular, angle \( y + x=90^\circ \). But we know \( x = 45^\circ \), so \( y=45^\circ \)? Wait, no, wait. Wait, the horizontal line is a straight line, so the sum of angles on one side of the horizontal line should be 180°? No, the vertical line (up) and horizontal line (left - right) form a right angle (90°). The angle between the upper vertical (up) and the line with 45° is 45°, so angle \( x = 45^\circ \). Then, the angle between the horizontal (left) and the line with angle \( y \) is equal to angle \( x \) because of the symmetry (the lines are symmetric with respect to the vertical line). So angle \( y = 45^\circ \)? Wait, no, let's do it properly. The horizontal line (left - right) and the vertical line (up - down) are perpendicular (90°). The angle between the upper vertical (up) and the line with 45° is 45°, so the angle between that line and the horizontal (right) is 45° (since 90 - 45 = 45). Then, the line making angle \( y \) with the horizontal (left) is symmetric to the line making 45° with the horizontal (right) with respect to the vertical line. So angle \( y = 45^\circ \). Alternatively, since the angle between the horizontal line (left) and the line with angle \( y \) and the vertical line (up) is such that angle \( y + x=90^\circ \), and \( x = 45^\circ \), so \( y = 45^\circ \).

Answer:

\( x = 45^\circ \), \( y = 45^\circ \)