91Ó°ÊÓ

Angles measure the rotation between two rays that share a common endpoint, known as the vertex. In everyday life and in math, angles are commonly measured using degrees. Think of a degree as a slice of a circle, where a full circle is comprised of 360°. This makes it helpful to conceptualize rotations — if you spin around completely you have turned 360°.
If you imagine starting from a point, moving in one direction, and then rotating until you reach the same direction, you'll have covered 360°. Angles less than this represent partial rotations, while multiples indicate full circles plus some extra rotation.

• A right angle, like the one at the corner of a piece of paper, is 90°.
• A straight angle, created by a straight line, is 180° because it is half of a full circle.
• A complete circle is 360°.

Angles can be either positive or negative, which indicates the direction of movement around the circle. A positive angle means you are rotating counterclockwise from the starting point. Conversely, a negative angle suggests a clockwise movement.
In the case of our 90° angle problem,

• Add 360° to create a positive coterminal angle: 450° (since adding moves in a counterclockwise direction).
• Subtract 360° to get a negative coterminal angle: -270° (subtracting indicates a clockwise movement).

This way, even if angles differ in value, they can eventually point in the same direction, creating what we call coterminal angles.

Rotational symmetry is evident when a shape or figure looks the same after some rotation (less than a full circle). With angles, especially in the context of coterminal angles, rotational symmetry reveals itself because angles that differ by full circle rotations appear identical. For example, starting at 90° and adding full 360° rotations, such as reaching 450°, results in a position indistinguishable from one another — hence they are coterminal due to this symmetry. The formula \[ 90^{\circ} + 360^{\circ}k \] exploits this concept to find all coterminal angles, where changing the integer \( k \) yields all potential symmetrical positions. Whether through rotation by positive angles (counterclockwise) or negative ones (clockwise), the end look remains constant.