I don't think two sides are known.

Yes, my statement was incorrect. However one side is known as well as all of the angles. So the solution can still proceed from there.

Using the notation in the attachment (I made this before seeing Skipjack's notation, so I used A, B and C for the angles):

Angles of triangle PQT are A, 90, 90-A.

Therefore, angles of RST are also A, 90, 90-A.

Side RS is known. So all other sides of RST can be determined.

Now, all sides of PQT can be determined.

Triangle sides PS and RS are known, so common hypotenuse PR can be determined.

Angles B and C can now be determined.