RESPONSES to PUZZLE #7
Very
few responses on this one, maybe that is good?
If
you are still working on it, keep working.
Here
is a hint:

As
for the reasoning, here is one approach.
Notice
that the sum of the measure of the interior angles
of the original quadrilateral is 360deg., since the sum of the
measure of the interior angles of a triangle is 180deg.?
Now
look at the point P and the corners of the quadrilateral
which meet at P.
See
anything interesting there?