Angles and their measure
In a triangle an angle never passes 180 degrees. What changes when an angle becomes an amount of turning, which can go around more than once, or backward?
Where you are 4 inside
What this area is about
Geometry measures an angle as a figure, between 0^\circ and 180^\circ. Trigonometry needs angles that turn: any amount, either direction, pinned to the coordinate plane. It also needs a unit that comes from the circle itself, the radian, which makes the formulas joining an angle to a length come out clean. A full turn is 360^\circ, or 2\pi radians.
How these fit together
Two definitions. Angles in standard position put the vertex at the origin and the initial side on the positive x-axis; counterclockwise is positive, clockwise negative, and coterminal angles share a terminal side, differing by whole turns. Radian measure measures an angle as arc over radius, \displaystyle \theta = \frac{s}{r}, and converts between radians and degrees, minutes and seconds.
What radians buy. Arc length and sector area in radians gives s = r\theta and \displaystyle A = \frac{1}{2}r^2\theta, which hold only with \theta in radians. Angular and linear speed separates how fast an object turns from how fast one point of it travels; with \omega in radians per unit of time, v = r\omega.
A path through it
Radian measure comes before arc length and angular speed, since both are built on it. Standard position is not a prerequisite of either, but radian measure and angular speed refer back to it, so it is a natural first lesson.
Next, these angles go onto the unit circle, where sine and cosine become coordinates, and into polar coordinates; the turning wheel model uses angular speed.