To determine which of the given measures represents the measures of all angles that are coterminal with a [tex]\(500^{\circ}\)[/tex] angle, let's follow these steps:
1. Understand Coterminal Angles: Coterminal angles are angles that share the same terminal side when in standard position. To find coterminal angles, you can add or subtract [tex]\(360^{\circ}\)[/tex] (a full rotation) any number of times ([tex]\(n\)[/tex]).
2. Simplify the Angle: Simplify [tex]\(500^{\circ}\)[/tex] by subtracting [tex]\(360^{\circ}\)[/tex] until the angle is within the range of [tex]\(0^{\circ}\)[/tex] to [tex]\(360^{\circ}\)[/tex].
[tex]\[ 500^{\circ} - 360^{\circ} = 140^{\circ} \][/tex]
Thus, [tex]\(500^{\circ}\)[/tex] is coterminal with [tex]\(140^{\circ}\)[/tex].
3. Identify the General Form: To find the measures of all angles coterminal with [tex]\(140^{\circ}\)[/tex], we can write:
[tex]\[ 140^{\circ} + 360n \][/tex]
Where [tex]\(n\)[/tex] is any integer (positive, negative, or zero), representing the number of complete rotations.
4. Compare with Given Options: Now, let's compare [tex]\(140^{\circ} + 360n\)[/tex] with the given options:
- [tex]\((40 + 360n)^\circ\)[/tex]
- [tex]\((140 + 360n)^\circ\)[/tex]
- [tex]\((220 + 360n)^\circ\)[/tex]
- [tex]\((320 + 360n)^\circ\)[/tex]
Clearly, the expression [tex]\(140 + 360n\)[/tex] exactly matches our simplified coterminal angle with the general form.
Thus, the measure that represents all angles coterminal with [tex]\(500^{\circ}\)[/tex] is:
[tex]\[ \boxed{(140 + 360n)^\circ} \][/tex]
