How do we know how an angle is defined? Take the angle PQR.  Below it's labeled 45 degrees.
How do we know PQR is 45 degrees and not 315 degrees (the reflexive angle)?  And I know it says "45 degrees", but what if it said 'X'.  Then how do we know PQR isn't '360-X' degrees?

 A: Short answer: we don't unless there's more information!
One convention that might be assumed in the context of the work you're doing is that an angle is the smallest positive angle between the rays $\vec{QR}$ and $\vec{QP}$, in which case it's $\frac{\pi}{4} = 45^\circ$.
Or perhaps the placement of the label on the diagram indicates the intended angle.

By the way, if you're looking for a definition of an angle, you have to draw in a circular arc with center at the vertex $Q$. Then the angle is defined as the ratio of the length of the arc to the radius of the circle:
$$
\theta = \frac{s}{r},
$$
which simplifies to $\theta = s$ if you draw the circular arc with radius $r=1$. This is of course, the natural unitless way to measure angle. If you insist on dividing a circle up into the (arbitrary) number of $360$ equal pieces then you have to convert:
$$
\theta^\circ = \theta \cdot \frac{360^\circ}{2\pi}.
$$
A: The angle $\angle PQR$ is measured by starting at the ray $\overrightarrow{QP}$ and then rotating that ray counterclockwise until we end up at the ray $\overrightarrow{QR}$. The direction matters. $\angle PQR$ and $\angle RQP$ are generally not the same. In fact, you measured the angle the wrong way round. In your example, the angle you drew is $\angle RQP$, not $\angle PQR$, and $\angle PQR$ would be $315^\circ$, not $45^\circ$.
