As title suggests, in the following quadrilaterals with some given angles and lengths, the goal is to find the measure of the base length $AB$. This is a contest-preparation problem with several different ways to solve it. I'll post my own approach below as an answer, please share your solutions as well!
In the quadrilateral $ABCD$, find the measure of length $AB$.
contest-matheuclidean-geometrygeometrytrigonometry
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Best Answer
This is my approach to this problem:
Extend $BC$ and $AD$ such that they intersect at point $E$. Since $\angle ECD=90$ and $\angle EDC=30$, we can conclude that $\angle AEB=60$, $\triangle AEB$ is equilateral and $\triangle CED$ is a $30-60-90$ triangle. Let $DE=2a$, this means that $CE=a$. Since $\triangle AEB$ is equilateral we can say that:
$$48+2a=60+a$$
$$a=12$$
Therefore, $x=60+a=60+12=72$