Each interior angle of a regular pentagon is $108$ degrees, whilst
each interior angle of a regular hexagon is $120$ degrees. The
non-regular pentagon in the centre of the diagram contains two
angles which are interior angles of the regular hexagon, two angles
which are interior angles of the regular hpentagon and a fifth
angle, the one marked $x$. So: $x+2\times120+2\times108=
5\times108=540$. Hence $x=84$.
This problem is taken from the UKMT Mathematical Challenges.