ABCD is a quadrilateral. Draw a diagonal AC, dividing ABCD into two triangles: ABC and ADC. The angles of a triangle add up to 180, so BAC+ABC+BCA=180 and DAC+ADC+DCA=180. Add these two equations together:
(BAC+DAC)+ABC+ADC+(BCA+DCA)=360 which is the same as BAD+ABC+ADC+BCD=360. These are the angles of the quadrilateral.