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I'm reading Conway's complex analysis book and on page 46 he said this:

What is the definition of a direction of an angle?

user42912
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An angle $BAC$ may be defined as the motion of one ray, say ray $AB$ here, in a definite direction (typically counterclockwise is taken as positive direction) toward the other ray, $AC$ here, where both rays are based at the same point $B$ and the ray is viewed as rotating around its initial point $B$.

With this definition, the angle $BAC$ will be the same as the negative of angle $CAB$ so that $CAB=-BAC.$

coffeemath
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This is related to the concept of orientation of a basis. In both cases it's not as important to define what orientation/direction is, as it is to define maps that preserve such things.

For the sake of clarity, let'slabel the two paths as $\gamma_1$ and $\gamma_2$. We can measure the angle between them in counterclockwise direction: by how much one needs to turn $\gamma_1$ counterclockwise so it becomes tangent to $\gamma_1$. Or, we could do it clockwise, getting the supplementary angle.

The author specifies that the counterclockwise angle between $\gamma_1$ and $\gamma_2$ must be equal to the counterclockwise angle between $f\circ \gamma_1$ and $f\circ \gamma_2$.