Question

find the measure of the missing angles.
diagram of intersecting lines with 59°, a right angle, and angles labeled x, y
answer attempt 2 out of 2
x = \boxed{^circ} quad y = \boxed{^circ}

Explanation:

Step1: Find \( x \)

We know that a right angle is \( 90^\circ \), and the sum of angles \( 59^\circ \), \( 90^\circ \), and \( x \) should be \( 180^\circ \) (since they form a straight line). Wait, actually, looking at the vertical angles or the linear pair? Wait, no, the angle with the right angle ( \( 90^\circ \) ), \( 59^\circ \), and \( x \): Wait, actually, the angle adjacent to \( 59^\circ \) and the right angle: Wait, let's correct. The angle \( x \), \( 59^\circ \), and the right angle ( \( 90^\circ \) ) are in a straight line? Wait, no, the straight line is \( 180^\circ \). So \( 59^\circ + 90^\circ + x = 180^\circ \)? Wait, no, maybe \( x \) and \( 59^\circ \) are complementary to the right angle? Wait, no, let's see: The right angle is \( 90^\circ \), so \( 59^\circ + x = 90^\circ \)? Wait, that makes sense because the angle between the two lines (one with \( 59^\circ \) and \( x \)) and the right angle. So \( x = 90^\circ - 59^\circ = 31^\circ \).
\[ x = 90 - 59 = 31 \]

Step2: Find \( y \)

Angle \( y \) and the \( 59^\circ \) angle: Wait, are they vertical angles? Wait, no, looking at the diagram, \( y \) and the \( 59^\circ \) angle: Wait, actually, \( y \) is equal to \( 59^\circ \)? Wait, no, wait, the right angle is \( 90^\circ \), and \( x = 31^\circ \), so \( y \) should be equal to \( 90^\circ - x \)? Wait, no, maybe \( y \) is equal to \( 59^\circ \) because of vertical angles? Wait, no, let's re-examine. The angle \( y \): The straight line is \( 180^\circ \), so \( 59^\circ + 90^\circ + y = 180^\circ \)? Wait, no, that would be \( y = 180 - 59 - 90 = 31 \), but that contradicts. Wait, no, maybe I made a mistake. Wait, the right angle is \( 90^\circ \), so the angle between the two lines (one with \( 59^\circ \) and the other with \( y \)): Wait, actually, \( x \) and \( y \): Wait, no, let's look at the vertical angles. The angle \( x \) and the angle opposite to \( y \)? Wait, no, the \( 59^\circ \) angle and \( y \): Wait, maybe \( y = 59^\circ \) because they are vertical angles? Wait, no, the right angle is \( 90^\circ \), so \( 59^\circ + x = 90^\circ \), so \( x = 31^\circ \), and then \( y = 90^\circ - x = 59^\circ \)? Wait, that makes sense. So \( y = 59^\circ \).
\[ y = 90 - 31 = 59 \] (Wait, no, earlier we had \( x = 31 \), so \( 90 - 31 = 59 \), so \( y = 59^\circ \). Alternatively, \( y \) is equal to \( 59^\circ \) because of the vertical angle or the complementary angle. So the correct values are \( x = 31^\circ \) and \( y = 59^\circ \).

Answer:

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