How can I define the exterior and interior angle of the figure?

Question

If i have this figure:

How to describe the angles, depending on the sides?, According to what I know, the angle $ K $, will be: the angle ABC, but with this, a problem arises, how do I describe the angle L? i think that can be CBA

Answer 1

If you want to specifically pick out the larger of the two angles, I don't think there is notation for that; you would need to explain in words. By default, $\angle ABC$ and $\angle CBA$ would both refer to the smaller angle.

You could use the convention of oriented angles, where we define $\angle ABC$ to be the angle of the counterclockwise rotation which takes ray $\overrightarrow{BA}$ to ray $\overrightarrow{BC}$. (Here, "counterclockwise" is an arbitrary choice, but mathematicians prefer counterclockwise angles to be positive.) This is well-defined up to adding a multiple of $2\pi$ (or $360^\circ$).

With this convention, $\angle ABC$ and $\angle CBA$ would be the two angles, but which one is which depends not on which angle is larger, but on which direction is the counterclockwise direction. In the diagram in the question, angle $K$ would be $\angle CBA$, since we rotate $\overrightarrow{BC}$ by an angle of $K$ counterclockwise to get $\overrightarrow{BA}$, and angle $L$ would be $\angle ABC$.

Using oriented angles, no matter which angle is which, we always have $\angle ABC = 360^\circ - \angle CBA$ (or just $\angle ABC = -\angle CBA$, since these are the same for oriented angles). Oriented angles also let us write equations like $\angle ABC + \angle CBD = \angle ABD$ without caring about the relative positions of points $A,B,C,D$.