Question

find the measure of the missing angles. 64° x y

Explanation:

Step1: Find angle \( x \)

We know that the angle of \( 64^\circ \), angle \( x \), and the right angle (\( 90^\circ \)) are related? Wait, no, actually, looking at the vertical angles or the straight line? Wait, the angle \( x \) and \( 64^\circ \) and the right angle: Wait, the sum of angles on a straight line? Wait, no, the angle \( x \), \( 64^\circ \), and the right angle (since there is a square, which is \( 90^\circ \))? Wait, no, let's re - examine. The angle \( x \) and \( 64^\circ \) are complementary to the right angle? Wait, no, the sum of \( x \), \( 64^\circ \), and \( 90^\circ \) should be \( 180^\circ \)? Wait, no, the angle \( x \) and \( 64^\circ \) are such that \( x + 64^\circ=90^\circ \)? Wait, no, the right angle is \( 90^\circ \), and the angle adjacent to \( 64^\circ \) and \( x \) is a right angle. So \( x + 64^\circ=90^\circ \), so \( x = 90^\circ - 64^\circ=26^\circ \).

Step2: Find angle \( y \)

Angle \( y \) is equal to angle \( x \) because they are vertical angles? Wait, no, angle \( y \) and the angle \( x \) and the right angle? Wait, no, angle \( y \) is equal to \( 64^\circ \)? Wait, no, let's see. The angle \( y \) and \( x \) and the right angle: Wait, the sum of angles around a point? No, the angle \( y \) is equal to \( 64^\circ \)? Wait, no, earlier we found \( x = 26^\circ \), and angle \( y \) is equal to \( x \)? Wait, no, maybe I made a mistake. Wait, the angle \( x \) and \( 64^\circ \) are complementary to \( 90^\circ \), so \( x=90 - 64 = 26^\circ \). Then angle \( y \) is equal to \( 64^\circ \)? Wait, no, angle \( y \) and \( x \) and the right angle: Wait, the sum of \( y \), \( x \), and \( 90^\circ \) is \( 180^\circ \)? No, the straight line is \( 180^\circ \). Wait, the angle \( y \) and the right angle and \( 64^\circ \): Wait, \( y+90^\circ + 64^\circ=180^\circ \)? No, that would be \( y = 180 - 90 - 64=26^\circ \)? Wait, no, I think I messed up. Wait, the angle \( x \) and \( 64^\circ \) are complementary (since they form a right angle), so \( x = 90 - 64=26^\circ \). Then angle \( y \) is equal to \( 64^\circ \)? Wait, no, angle \( y \) is equal to \( x \)? Wait, no, let's look at the vertical angles. The angle \( y \) and the angle \( 64^\circ \) are not vertical angles. Wait, the angle \( y \) and the angle \( x \) are such that \( y=x = 26^\circ \)? No, that can't be. Wait, no, the right angle is \( 90^\circ \), and the angle \( y \) and \( 64^\circ \) are such that \( y + 64^\circ=90^\circ \)? No, I think I made a mistake in the first step. Let's start over.

The angle with the square is a right angle (\( 90^\circ \)). The angle \( x \), \( 64^\circ \), and the right angle are on a straight line, so their sum is \( 180^\circ \). So \( x+64^\circ + 90^\circ=180^\circ \), so \( x=180 - 90 - 64 = 26^\circ \). Then angle \( y \) is equal to \( x \) because they are vertical angles? Wait, no, angle \( y \) is equal to \( 64^\circ \)? Wait, no, angle \( y \) and \( x \) are vertical angles? No, angle \( y \) and the angle \( 64^\circ \) are vertical angles? Wait, no, the vertical angle of \( x \) is \( y \)? Wait, no, the two lines intersect, so the vertical angle of \( x \) is \( y \), so \( y=x = 26^\circ \)? No, that's not right. Wait, I think the correct approach is: The angle \( x \) and \( 64^\circ \) are complementary (since they form a right angle), so \( x = 90 - 64=26^\circ \). Then angle \( y \) is equal to \( 64^\circ \) because they are vertical angles? Wait, no, angle \( y \) is equal to \( x \)? I'm confused. Wait, let's use the fact that the sum of angles on a straight line is \( 180^\circ \). The straight line has angles \( 64^\circ \), \( 90^\circ \), and \( y \)? No, the straight line has angles \( x \), \( 64^\circ \), and \( 90^\circ \), so \( x + 64+90 = 180\), so \( x=26 \). Then angle \( y \) is equal to \( x \) because they are vertical angles, so \( y = 26^\circ \)? No, that can't be. Wait, maybe the right angle is between \( y \) and \( 64^\circ \), so \( y+64 = 90\), so \( y = 26^\circ \), and \( x = 64^\circ \)? I think I mixed up \( x \) and \( y \). Let's re - label. The angle with \( 64^\circ \), the right angle, and \( y \): \( 64 + 90+y=180\), so \( y=180 - 64 - 90 = 26^\circ \). And \( x \) is equal to \( 64^\circ \) because they are vertical angles. Wait, now I see. The angle \( x \) and \( 64^\circ \) are vertical angles, so \( x = 64^\circ \), and \( y=26^\circ \). Oh! I made a mistake earlier. The vertical angle of \( 64^\circ \) is \( x \), so \( x = 64^\circ \), and the vertical angle of \( y \) is the angle \( 26^\circ \) (since \( 64 + 90+y=180\), \( y = 26^\circ \)).

Wait, let's correct:

Step1: Find \( x \)

Angle \( x \) and the \( 64^\circ \) angle are vertical angles, so \( x = 64^\circ \).

Step2: Find \( y \)

The sum of \( 64^\circ \), \( 90^\circ \), and \( y \) is \( 180^\circ \) (since they are on a straight line). So \( 64^\circ+90^\circ + y=180^\circ \).

\( y=180^\circ-(64^\circ + 90^\circ)=180^\circ - 154^\circ = 26^\circ \).

Answer:

\( x = 64^\circ \), \( y = 26^\circ \)