In the adjoining figure, AP and EQ are respectively the bisectors of $$\angle BAC$$ and $$\angle DEF$$. Then, the measure of angle $$x$$ is

In the triangle $$ABC$$, the angle $$\angle ACB$$ is vertically opposite the marked $$70^\circ$$, so $$\angle BAC = 180^\circ - 30^\circ - 70^\circ = 80^\circ$$ and the bisector $$AP$$ splits it into two angles of $$40^\circ$$. Where $$AP$$ crosses the line $$BD$$, the angle formed with $$BD$$ is $$180^\circ - 30^\circ - 40^\circ = 110^\circ$$, that is $$70^\circ$$ on the other side. In the four sided figure $$CDEF$$ the angles at $$D$$ and at $$F$$ are right angles and the angle at $$C$$ is $$70^\circ$$, so $$\angle DEF = 360^\circ - 70^\circ - 90^\circ - 90^\circ = 110^\circ$$, and its bisector $$EQ$$ makes $$55^\circ$$ with $$ED$$ and therefore $$35^\circ$$ with $$BD$$. The two bisectors together with $$BD$$ form a triangle, so the angle between the bisectors is $$180^\circ - 70^\circ - 35^\circ = 75^\circ$$, and $$x$$ is vertically opposite that angle.