Answer:
Step 1: Given AD || BC and AB || CD.
Step 2: Angle A and Angle C are alternate interior angles, so they are congruent.
Step 3: Angle B and Angle D are alternate interior angles, so they are congruent.
Step 4: Angle B + Angle D = 180 degrees (because Angle A + Angle B + Angle C + Angle D = 360 degrees for a quadrilateral)
Step 5: Since Angle B and Angle D are congruent and their sum is 180 degrees, they must each be 90 degrees.
Step 6: Triangle ABZ and Triangle CDZ are both right triangles, since Angle B and Angle D are each 90 degrees.
Step 7: By the Angle-Angle Similarity Theorem, triangles ABZ and CDZ are similar.
Step 8: Since triangles ABZ and CDZ are similar, corresponding sides are proportional.
Step 9: Therefore, ZB/ZD = AB/CD, since ZB corresponds to AB and ZD corresponds to CD.
Step 10: Given that AB || CD, we know that AB/CD = AZ/CZ (the corresponding sides of similar triangles are proportional).
Step 11: Therefore, ZB/ZD = AB/CD = AZ/CZ.
Step 12: Since Angle A and Angle C are congruent (Step 2), we know that AZ = CZ (since they are opposite sides of an isosceles triangle).
Step 13: Therefore, ZB/ZD = AZ/CZ = 1, which means that ZB = ZD.
Thus, we have proved that ZB = ZD!
- Dante