Answer:
A missing bit of information is that ABCD must be a rectangle.
First, let's gather some facts:
(1) Angles in a triangle add up to 180.
(2) RQC and PBQ are the same triangles since it is given that their sides are equal.
(3) Also ∠PBQ = ∠QCR are 90° since they are in fact the corners of the rectangle.
(4) ∠PQB, ∠PQR and∠RQC are on the same line, so their sum must be 180°
Putting it all together, we can say for both triangles using (1):
∠RQC + ∠QRC + ∠QCR = 180 and ∠QCR=90 => ∠RQC = 90-∠QRC
∠PBQ + ∠BQP + ∠PQB = 180
Furthermore, since both triangles are equal (2):
∠QRC = ∠PQB
So, ∠RQC = 90 - ∠PQB
Finally, using (4) we can say:
∠PQB + ∠PQR + 90 - ∠PQB = 180.
Simplify that and you get:
∠PQR + 90 = 180
∠PQR = 90