Question

Suppose a triangle has sides a, b, and c, and let theta be opposite the side of length a. If cos theta < 0, what must be true?

Answer 1

If $\cos \theta\ \textless \ 0$, then angle $\theta$ is obtuse and triangle with sides a, b, c is obtuse triangle.
In an arbitrary triangle can be only one obtuse angle, and the side which lies opposite to the largest angle is the largest. Then since angle $\theta$ is opposite the side of length a  you can conclude that a>c and a>b.