Question

find the measure of the missing angles.

Explanation:

Step1: Find the measure of 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 and the given \( 55^\circ \) angle) can be used. The angle \( x \), \( 55^\circ \), and the right angle (\( 90^\circ \)) are related such that \( x + 55^\circ + 90^\circ = 180^\circ \) (since they form a straight line). Wait, actually, looking at the diagram, the angle \( x \) and \( 55^\circ \) are complementary to the right angle? Wait, no. Let's re-examine. The right angle is \( 90^\circ \), and the angle between the \( 55^\circ \) angle and \( x \) is a right angle? Wait, no, the diagram shows that there is a right angle (the square) between two lines, and a \( 55^\circ \) angle, and angle \( x \). So, for angle \( x \): since \( 55^\circ + x + 90^\circ = 180^\circ \)? Wait, no, actually, the sum of angles on a straight line is \( 180^\circ \). So, \( 55^\circ + x + 90^\circ = 180^\circ \)? Wait, no, that would be if they are on a straight line. Wait, maybe \( x + 55^\circ = 90^\circ \), because the right angle is \( 90^\circ \), so \( x = 90^\circ - 55^\circ \). Let's calculate that: \( x = 90 - 55 = 35^\circ \).

Step2: Find the measure of angle \( y \)

Angle \( y \) is adjacent to the right angle and the \( 55^\circ \) angle? Wait, no. Wait, angle \( y \) and the right angle and the \( 55^\circ \) angle? Wait, no, looking at the diagram, angle \( y \) is formed with the right angle and the vertical (or the other line). Wait, actually, angle \( y \) is equal to \( 90^\circ + 55^\circ \)? Wait, no. Wait, the sum of angles around a point? No, let's see. The straight line: angle \( y \) and the angle opposite to \( x + 55^\circ \)? Wait, no. Wait, angle \( y \) is adjacent to the right angle and the angle that is equal to \( x \)? No, maybe angle \( y \) is \( 90^\circ + 55^\circ \)? Wait, no, let's think again. The right angle is \( 90^\circ \), and the angle \( y \) is formed with the line that has the \( 55^\circ \) angle. Wait, actually, angle \( y \) is \( 90^\circ + 55^\circ = 145^\circ \)? Wait, no, that can't be. Wait, no, the sum of angles on a straight line: angle \( y \) and the angle that is \( x \) (which is \( 35^\circ \))? No, wait, maybe angle \( y \) is \( 180^\circ - 35^\circ \)? No, that doesn't make sense. Wait, let's start over.

Wait, the diagram: two lines intersect, forming a right angle (90 degrees), a 55-degree angle, angle x, and angle y. Let's identify the angles. The angle between the 55-degree angle and angle x is 90 degrees (the right angle). So, 55 + x = 90, so x = 35 degrees (as in Step 1). Then, angle y: since angle y and the angle that is 55 degrees are supplementary? No, wait, angle y is adjacent to the right angle and the angle that is equal to x? Wait, no, angle y is formed with the line that has the right angle and the line that has the 55-degree angle. Wait, actually, angle y is equal to 90 + 55 = 145? No, that's not right. Wait, no, the sum of angles on a straight line: angle y and the angle that is x (35 degrees) and the right angle? No, maybe angle y is 180 - 35 = 145? Wait, no, let's see. The straight line: angle y and the angle opposite to x? No, maybe angle y is 90 + 55 = 145? Wait, let's check. If x is 35, then the angle opposite to x is also 35 (vertical angles). Then, angle y is adjacent to the right angle (90) and the 55-degree angle? Wait, no, the right angle is 90, the 55-degree angle, and angle y: wait, maybe angle y is 90 + 55 = 145. Let's confirm. The sum of angles around a point: but maybe the straight line. Wait, the line with angle y and the angle that is 35 degrees (x) and the right angle? No, I think I made a mistake. Wait, let's use vertical angles and linear pairs.

Wait, the angle x is 35 degrees (as 90 - 55 = 35). Then, angle y: since angle y and the angle that is 35 degrees (x) and the right angle? No, angle y is adjacent to the right angle (90) and the angle that is 55 degrees? Wait, no, the right angle is 90, the 55-degree angle, and angle y: if we look at the straight line, the sum of angles on a straight line is 180. So, angle y + 35 = 180? No, that would be if angle y and x are supplementary. Wait, no, x is 35, so angle y is 180 - 35 = 145? Wait, no, that's not. Wait, maybe angle y is 90 + 55 = 145. Let's calculate: 90 + 55 = 145. Yes, that makes sense. Because the right angle is 90, and the 55-degree angle is adjacent to it, so angle y is 90 + 55 = 145 degrees.

Wait, let's verify:

For angle x:
The right angle (90°) and the 55° angle and angle x are on a straight line? No, wait, the right angle is between two lines, and the 55° angle and angle x are on one side of the right angle. So, 55° + x = 90°, so x = 35°. That's correct because they are complementary (sum to 90°).

For angle y:
Angle y is adjacent to the right angle and the 55° angle? Wait, no, angle y is formed by the line that has the right angle and the line that has the 55° angle. Wait, actually, angle y is equal to 90° + 55° = 145°, because the right angle (90°) and the 55° angle are adjacent to angle y, forming a linear pair? No, wait, the sum of angle y and the angle opposite to x (which is 35°) would be 180°, but angle y is 180° - 35° = 145°, which matches 90° + 55° = 145°. So that's correct.

So, angle x is 35°, angle y is 145°.

Answer:

Angle \( x = 35^\circ \), Angle \( y = 145^\circ \)