How to find angle of vertex given you know all the side lengths of a scalene triangle? Let's say I know all the side lengths of a scalene triangle. How would I find the angle of a particular vertex given this information?

Given side lengths $a,b$ and $c$, you can use cosine rule to easily find all the angles.

$$\cos \mathrm A= \frac{b^2+c^2-a^2}{2bc}$$

Similarly, $$\cos \mathrm B=\frac{a^2+c^2-b^2}{2ac}$$

$$\cos \mathrm C=\frac{a^2+b^2-c^2}{2ab}$$