The key to this proof lies in the assertion PQ=PS and RQ=RS.  With these two relationships, we can envision a standard kite where the bottom portion is longer than the top portion.  Let's say that P is the top point of the kite, R is the bottom point of the kite, Q is the left point of the kite, and S is the right point of the kite. This gives us two basic facts --

* The line segment running from the top point (P) to the left point (Q) is the same length the line segment running from the top point (P) to the right point (S).

* The line segment running from the bottom point (R) to the left point (Q) is the same length as the line segment running from the bottom point (R) to the right point (S).

Because line segments PQ and PS are the same length (PQ=PS), and line segments RQ and RS are the same length (RQ=RS), the kite is symmetrical.  Specifically, you can fold the kite in half along the line segment PR and the left side of the kite (the triangle PQR) will lay exactly on top of the right side (the triangle PSR).

You can use these facts to prove statements a-d by contradiction.  If the kite wasn't symmetrical, the lengths of PQ and PS would be different, or the lengths of RQ and RS would be different, which contradicts the original assertion (PQ=PS and RQ=RS).