As before consider quadrilateral ABCD (Fig 3.6). Let P be any point in its interior. Join P to vertices A, B, C and D. In the figure, consider \Delta PAB. From this we see x = 180\degree - m\angle 2 - m\angle 3; similarly from \Delta PBC, y = 180\degree - m\angle 4 - m\angle 5, from \Delta PCD, z = 180\degree - m\angle 6 - m\angle 7 and from \Delta PDA, w = 180\degree - m\angle 8 - m\angle 1. Use this to find the total measure m\angle 1 + m\angle 2 + ... + m\angle 8, does it help you to arrive at the result? Remember \angle x + \angle y + \angle z + \angle w = 360\degree.
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