Proof: If Angles Congruent Trapezoid Isosceles

Let's prove the following theorem:

if m∠ZWX = m∠YXW and WX || ZY, then distance WZ = distance XY

P W X Y Z

Proof:

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Given
1 m∠ZWX = m∠YXW
2 WX || ZY
Assumptions
3 YP || ZW
4 m∠XPW = 180
Proof Table
# Claim Reason
1 m∠WPX = 180 if m∠XPW = 180, then m∠WPX = 180
2 m∠ZWP = m∠YXP if m∠ZWX = m∠YXW and m∠WPX = 180, then m∠ZWP = m∠YXP
3 m∠YPX = m∠ZWP if YP || ZW and m∠XPW = 180, then m∠YPX = m∠ZWP
4 m∠YPX = m∠YXP if m∠YPX = m∠ZWP and m∠ZWP = m∠YXP, then m∠YPX = m∠YXP
5 distance YP = distance YX if m∠YPX = m∠YXP, then distance YP = distance YX
6 WP || ZY if WX || ZY and m∠WPX = 180, then WP || ZY
7 WPYZ is a parallelogram if WP || ZY and YP || ZW, then WPYZ is a parallelogram
8 distance WZ = distance PY if WPYZ is a parallelogram, then distance WZ = distance PY
9 distance PY = distance XY if distance YP = distance YX, then distance PY = distance XY
10 distance WZ = distance XY if distance WZ = distance PY and distance PY = distance XY, then distance WZ = distance XY

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