One way to think about it is this: the longer base and the angles determine a triangle. The height determines where the similar triangle is cut off to give the trapezium. Thus, if we let the base you want be $b,$ and the other $B,$ then it follows that $$\frac bB=\frac aA.$$
You can determine $A,$ the height of the main triangle, from the given information. Then $a,$ the height of the cut-off triangle, is given by $a=A-h,$ with $h$ being the height of the trapezium, also given.
The work is in determining $A.$ One way is to first calculate some other side of the large triangle (use the cosine rule, for example -- remember all three angles are known, and one side). Then you now have two sides and an included angle. This allows you to determine $A$ as a scaled sine of the included angle.