Question

angle x is coterminal with angle y. if the measure of angle x is greater than the measure of angle y, which statement is true regarding the values of x and y?
○ x = y - 180n, for any positive integer n
○ x = y - 360n, for any integer n
○ x = y + 360n, for any positive integer n
○ x = y + 180n, for any integer n

Explanation:

Step1: Recall Coterminal Angles Definition

Coterminal angles are angles that share the same terminal side. The formula for coterminal angles is that if two angles \( x \) and \( y \) are coterminal, then \( x = y + 360n \) or \( x=y - 360n \), where \( n \) is any integer (positive, negative, or zero). This is because adding or subtracting a full rotation (360 degrees) multiple times (represented by \( n \)) doesn't change the terminal side of the angle.

Step2: Analyze Each Option

• Option 1: \( x=y - 180n \). 180 degrees is a half - rotation, not a full rotation. Angles that differ by 180n are supplementary or have a relationship related to straight angles, not coterminal. So this option is incorrect.
• Option 2: \( x=y - 380n \). 380 is not a multiple of 360. The correct multiple for coterminal angles is 360 (the measure of a full rotation), so this option is incorrect.
• Option 3: \( x=y + 360n \), for any positive integer \( n \). The formula for coterminal angles allows \( n \) to be any integer (including negative integers, which would correspond to subtracting 360 degrees multiple times). But among the given options, this is the closest to the correct formula. Since \( x \) is coterminal with \( y \) and \( x>y \), we can write \( x=y + 360n \) where \( n \) is a positive integer (because if \( n \) were negative, \( x \) would be less than \( y \)).
• Option 4: \( x=y + 180n \). Again, 180 is a half - rotation, not a full rotation, so angles related by 180n are not coterminal.

Answer:

C. \( x = y+360n \), for any positive integer \( n \) (assuming the third option in the original problem is labeled as C, with the text "x = y + 360n, for any positive integer n")