Question

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

Explanation:

Step1: Find x using vertical angles

Vertical angles are equal, so \( x = 59^\circ \) (since the angle opposite to \( 59^\circ \) is \( x \)).

Step2: Find y using angle sum (right angle)

We know there's a right angle (\( 90^\circ \)) and \( 59^\circ \), \( x \), and \( y \) around a point? Wait, actually, the right angle, \( 59^\circ \), and \( y \) form a straight line? Wait, no, looking at the diagram, the right angle (square) means \( 90^\circ \), and we have \( 59^\circ \), \( 90^\circ \), and \( y \) in a straight line? Wait, no, let's re - examine. The angles around a point? Wait, actually, the angle with the square is \( 90^\circ \), and we have \( 59^\circ \), \( 90^\circ \), and \( y \) such that \( 59^\circ+90^\circ + y=180^\circ \)? Wait, no, maybe the straight line. Wait, the angle \( x = 59^\circ \) (vertical angles), and then for \( y \), we know that \( 59^\circ+90^\circ + y = 180^\circ \)? Wait, no, let's think again. The right angle is \( 90^\circ \), the angle of \( 59^\circ \), and \( y \) are in a straight line (supplementary angles). Wait, the sum of angles on a straight line is \( 180^\circ \). So \( 59^\circ+90^\circ + y=180^\circ \)? Wait, no, that would be if they are on a straight line. Wait, actually, the angle \( x \) and the \( 59^\circ \) angle are vertical angles, so \( x = 59^\circ \). Then, for \( y \), we know that \( 59^\circ+90^\circ + y=180^\circ \)? Wait, no, the right angle is \( 90^\circ \), and we have \( 59^\circ \), \( 90^\circ \), and \( y \) such that \( 59 + 90+y=180 \)? Wait, \( 59+90 = 149 \), then \( y=180 - 149=31 \)? Wait, no, maybe the right angle, \( y \), and the angle equal to \( x \) (which is \( 59^\circ \))? Wait, no, let's correct. The right angle is \( 90^\circ \), and we have three angles: \( 59^\circ \), \( 90^\circ \), and \( y \) that add up to \( 180^\circ \) (since they are on a straight line). So \( 59 + 90+y=180 \). So \( y=180-(59 + 90)=180 - 149 = 31^\circ \). And \( x \) is vertical to \( 59^\circ \), so \( x = 59^\circ \).

Wait, let's re - do:

Step1: Determine x

Vertical angles are congruent. The angle labeled \( 59^\circ \) and angle \( x \) are vertical angles. So \( x=59^\circ \).

Step2: Determine y

We know that the sum of angles on a straight line is \( 180^\circ \). We have a right angle (\( 90^\circ \)), the \( 59^\circ \) angle, and angle \( y \) on a straight line. So \( 59^\circ+90^\circ + y = 180^\circ \).

First, calculate \( 59 + 90=149 \). Then, \( y=180 - 149 = 31^\circ \).

Answer:

\( x = 59^\circ \), \( y = 31^\circ \)