How can I tell whether a drawn angle represents a turn clockwise or counterclockwise?

Look for a curved arc with an arrowhead. The ray where the turn begins is the initial side; the ray where it ends is the terminal side. Follow the arrow from the initial side to the terminal side. If it travels in the direction of clock hands, it is clockwise; if it travels the opposite way, it is counterclockwise.

The two rays alone may not tell you which one is the start or which way the turn goes. For instance, a ray pointing right and a ray pointing up could show a turn from right to up, or a turn from up to right. Check for labels, an arrow, or instructions that identify the starting ray and direction.

The cited precalculus text defines counterclockwise rotation from initial side to terminal side as positive, and clockwise rotation as negative. That is the sign convention used in that text. [S1]

Suppose the initial ray points right and the terminal ray points up. The arc and arrow should show the direction of rotation around the vertex, like this schematic:

        terminal
           ↑
       ↗   |
initial →  •

The diagonal arrow is only a reminder of the direction; it is not a curved arc. On a clear diagram, draw an arc around the vertex with an arrowhead pointing along the turn. From the right-pointing ray to the upward ray along the shorter route is counterclockwise, so it is positive under the convention in the cited text. [S1]

If you travel around the other way from the same starting ray to the same ending ray, the turn is clockwise and follows a different route. The start and the path shown by the arrow matter—not just the two rays.

An arc can mark the region between the rays without making the direction of rotation clear. An angle size gives an amount, but it does not by itself identify which ray is the start or which way to turn. Look for other labels or wording in the problem before deciding.

For example, instructions might identify one ray as the initial side and specify a clockwise or counterclockwise turn. If the information is missing, ask which ray is the start and which direction is intended rather than guessing.

A diagram in standard position uses the positive x-axis as its initial side. That is a convention described in the cited text; do not assume it applies to every angle drawing. [S1]

Label the initial ray, label the terminal ray, and draw a curved arrow showing the rotation from the first to the second. If the sign matters, state the sign convention you are using. You can write:

“Start at ray A. Turn [clockwise/counterclockwise] through [amount] to ray B. Under this convention, the angle is [negative/positive].”

For example: “Start at ray A. Turn counterclockwise through 45° to ray B. Under this convention, the angle is positive.” This specifies the direction, amount, and sign convention rather than relying on the shape of the angle alone. The cited text uses positive for counterclockwise and negative for clockwise. [S1]