In the adjoining figure, $$ABC$$ is a triangle in which $$\angle BAC = 100^\circ$$, $$\angle ACB = 30^\circ$$. An equilateral triangle, a square and a regular hexagon are drawn as shown in the figure. The measure (in degrees) of $$(x + y + z)$$ is

The third angle of the triangle is $$180^\circ - 100^\circ - 30^\circ = 50^\circ$$, so the three angles of the triangle add to $$180^\circ$$. At each vertex of the triangle the angles round that point add to $$360^\circ$$, and the polygons contribute their interior angles $$60^\circ$$, $$90^\circ$$ and $$120^\circ$$, each at two vertices. Hence $$x + y + z = 3 \times 360^\circ - 180^\circ - 2(60^\circ + 90^\circ + 120^\circ) = 1080^\circ - 180^\circ - 540^\circ = 360^\circ$$.