Question

find the measure of the missing angles.

Explanation:

Step1: Find angle \( x \)

We know that a right angle is \( 90^\circ \), and the sum of angles in a straight line (or around a point with a right angle involved) can be used. Notice that \( x \), \( 51^\circ \), and the right angle (\( 90^\circ \)) are related such that \( x + 51^\circ+ 90^\circ = 180^\circ \) (since they form a straight line, sum to \( 180^\circ \)). Wait, actually, looking at the diagram, \( x \) and \( 51^\circ \) are complementary to the right angle? Wait, no, let's re-examine. The right angle is \( 90^\circ \), so \( x + 51^\circ = 90^\circ \) (because they are adjacent to the right angle and form a right angle together). So \( x = 90^\circ - 51^\circ \).
\( x = 90 - 51 = 39^\circ \)

Step2: Find angle \( y \)

Now, angle \( y \), the right angle (\( 90^\circ \)), and \( 51^\circ \) – wait, actually, angle \( y \) and \( x \) are vertical angles? No, wait, let's see. Alternatively, since \( y \) and the angle adjacent to it (which is \( 51^\circ \)) and the right angle: Wait, actually, \( y \) should be equal to \( 51^\circ \)? No, wait, no. Wait, the angle opposite to \( 51^\circ \) – no, let's use the fact that vertical angles are equal, and also the right angle. Wait, actually, \( y \) and the angle that is \( 51^\circ \) – wait, no, let's look at the straight line. Wait, another approach: The angle \( y \), the right angle (\( 90^\circ \)), and \( x \) (which we found as \( 39^\circ \)) – wait, no, maybe \( y = 51^\circ \)? No, that's not right. Wait, no, let's see: The angle between the two lines, one with \( x \) and \( 51^\circ \), and the other with \( y \) and the right angle. Wait, actually, \( y \) and \( 51^\circ \) are vertical angles? No, wait, \( x = 39^\circ \), and then \( y \) is equal to \( 51^\circ \)? Wait, no, let's check the sum. Wait, the three angles \( x \), \( 51^\circ \), and \( 90^\circ \) – no, the straight line is \( 180^\circ \), so \( x + 90^\circ + y = 180^\circ \)? Wait, no, the diagram has a right angle, so the angle between the two lines is \( 90^\circ \). So, \( x + 51^\circ = 90^\circ \) (so \( x = 39^\circ \)), and then \( y \) is equal to \( 51^\circ \)? Wait, no, maybe \( y = 90^\circ - x \)? Wait, no, let's think again. The angle \( y \) and \( x \) and the right angle: Wait, the angle \( y \) is adjacent to \( x \) and the right angle? No, perhaps \( y = 51^\circ \). Wait, no, let's use vertical angles. The angle opposite to \( 51^\circ \) would be \( y \)? No, wait, the angle \( y \) and the angle with \( 51^\circ \) – maybe I made a mistake. Wait, let's start over.

Looking at the diagram: There is a right angle (marked with a square), so that's \( 90^\circ \). Then, we have an angle of \( 51^\circ \), angle \( x \), and angle \( y \). The sum of angles around a point is \( 360^\circ \), but we can look at the straight lines. The line with \( x \), \( 51^\circ \), and the right angle: Wait, the straight line (180 degrees) includes \( x \), \( 51^\circ \), and the right angle? No, the right angle is \( 90^\circ \), so \( x + 51^\circ + 90^\circ = 180^\circ \)? Wait, that would mean \( x + 141^\circ = 180^\circ \), so \( x = 39^\circ \), which matches the first step. Then, the other straight line: \( y + 90^\circ + x = 180^\circ \)? Wait, \( y + 90^\circ + 39^\circ = 180^\circ \), so \( y = 180 - 90 - 39 = 51^\circ \). Ah, there we go. So \( y = 51^\circ \), \( x = 39^\circ \).

Answer:

\( x = 39^\circ \), \( y = 51^\circ \)