Question 13
In quadrilateral $ABCD,$ $\angle BAD \cong \angle ADC$ and $\angle ABD \cong \angle BCD,$ $AB = 8,$ $BD = 10,$ and $BC = 6.$ The length $CD$ may be written in the form $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$