Construct equilateral triangle ABW. Now:

∠WAY = 40° (because each interior angle of ABW is 60° and the interior angles of triangle XAY must total 180°)

p = p (because triangle ATY is isosceles)

q = q (because AB = AW = XY = p + q)

∠AWY = 40° (because triangles ATX and YTW are similar)

r = r (because triangle AYW is isosceles)

This shows that ABWY is a kite, a quadrilateral with two pairs of equal-length sides, each pair adjacent. The diagonals of a kite are perpendicular, so the triangle that contains angle z also contains interior angles measuring 90° and 80°. Angle z is 10°.

(Thanks, Derek.)