Since ABCD is a square, there are two things that you know already:
1) The angles DAB, ABC, BCD, CDA are all equal, and they're all right angles (ninety degrees).
2) The lines AB, BC, CD, DA, are all equal in length.
Now if AP = BQ = CR = DS, then PB = QC = RD = SA
So - given that two triangles with comparable sides and the angle contained between these two sides are equal (in this case, they're all right angles), that means that the following triangles are all congruent:
SAP is congruent to PBQ is congruent to QCR is congruent to RDS.
This means that the angles ASP, BPQ, CQR, and DRS are all equal. This also means that the angles APS, BQP, CRQ, and DSR are all equal.
Now, since all of these triangles are also right triangles (one angle is 90 degrees, a right angle), and since the angles in any triangle must sum to 180 degrees, this means that the other two angles in each of these four triangles must sum to 90 degrees:
Angles ASP + APS = BPQ + BQP = CQR + CRQ = DRS + DSR = 90 degrees.
But (look back at the congruent triangles) this also means that for the angles
APS + BPQ = BQP + CQR = CRQ + DRS = DSR + ASP = 90 degrees.
Given that the total of all angles at a point on a straight line is 180 degrees, that means that all of the following angles are 90 degrees (or, that they are right angles):
PQR = QRS = RSP = SPQ = 90 degrees.
Now, remember, that all of those triangles were congruent, so the following sides are all equal:
PQ = QR = RS = SP.
So, with all the sides equal, and all the angles being 90 degrees, PQRS is a square.
