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?

If , then angle 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 is opposite the side of length a  you can conclude that a>c and a>b.