Question

I need help with this. A picture for #8 is above.

Answer 1

Since the lengths of the segments AD=PD are equal that means that the triangle is an isosceles triangle. This also means that the angles A and P (in the triangle) are equal.

This also means that the angle P' (the one inside the quadrilateral) is equal to:

$180-\measuredangle B$

Now, if we extend the sides DP and CB we notice that the angles P' and Q

are equal

The only way that this angles can be equal is if the line segments DP and CB are parallel, this comes from the fact that two alternate interior angles are equal if and only if the lines are parallel.