jmc

By the Pythagorean Theorem, $$B{P}^2=A{P}^2-A{B}^2=20^2-16^2=144$$and so $BP=12$, since $BP>0$. Therefore, since $PT=BP$, $PT=12$.

By the Pythagorean Theorem, $$T{Q}^2=P{Q}^2-P{T}^2=15^2-12^2=81$$and so $TQ=9$, since $TQ>0$.

In triangles $PQA$ and $TQP$, the ratios of corresponding side lengths are equal. That is, $$\frac{PA}{TP}=\frac{PQ}{TQ}=\frac{QA}{QP}$$or $$\frac{20}{12}=\frac{15}{9}=\frac{25}{15}=\frac{5}{3}.$$Therefore, $△PQA$ and $△TQP$ are similar triangles and thus their corresponding angles are equal. That is, $\angle PQA=\angle TQP=\alpha$.

Since $\angle RQD$ and $\angle PQA$ are vertically opposite angles, then $\angle RQD=\angle PQA=\alpha$.

Since $CD$ and $TS$ are parallel, then by the Parallel Lines Theorem $\angle RDQ=\angle TQP=\alpha$.

Therefore, $\angle RDQ=\angle RQD$ and so $△RQD$ is an isosceles triangle with $QR=RD$, so $QR-RD=\boxed{0}$.