Folks, the question does not assume the shape of the right triangle in the semicircle. Thus, it can be assumed that the answer is INDEPENDENT of the shape. So, we are free to pick the shape that makes the figuring as easy as possible, for example with the apex of the triangle at the midpoint of the semicircle.

Now, when the perpedniculars are extend to the center of the inner circle, we get the angle between them as 135, 90, 135. the other angles of the two isosceles triangles are, thus, (180-135)/2. Combined at the "?" they yield 45 degrees.

This is a special case of the theorem that the central angle of two points on a circle is twice the angle of those points through a third point on the circle (on the larger arc; for a point on the shorter arc the angel is 180-45=135. The other two angle of the quadrilateral also total 180 so that the four total 360, as required for any quadrilateral!)