Proof: If Bisector Then Parallelogram

Let's prove the following theorem:

if distance WP = distance PY and distance XP = distance PZ and m∠WPY = 180 and m∠XPZ = 180, then WXYZ is a parallelogram

Z W X Y P

Proof:

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Given
1 distance WP = distance PY
2 distance XP = distance PZ
3 m∠WPY = 180
4 m∠XPZ = 180
Proof Table
# Claim Reason
1 m∠WPX = m∠YPZ if m∠WPY = 180 and m∠XPZ = 180, then m∠WPX = m∠YPZ
2 distance WP = distance YP if distance WP = distance PY, then distance WP = distance YP
3 distance PX = distance PZ if distance XP = distance PZ, then distance PX = distance PZ
4 △WPX ≅ △YPZ if distance WP = distance YP and m∠WPX = m∠YPZ and distance PX = distance PZ, then △WPX ≅ △YPZ
5 distance WX = distance YZ if △WPX ≅ △YPZ, then distance WX = distance YZ
6 m∠XWP = m∠ZYP if △WPX ≅ △YPZ, then m∠XWP = m∠ZYP
7 m∠ZYP = m∠PWX if m∠XWP = m∠ZYP, then m∠ZYP = m∠PWX
8 m∠ZYP = m∠ZYW if m∠WPY = 180, then m∠ZYP = m∠ZYW
9 m∠ZYW = m∠PWX if m∠ZYP = m∠ZYW and m∠ZYP = m∠PWX, then m∠ZYW = m∠PWX
10 m∠PWX = m∠YWX if m∠WPY = 180, then m∠PWX = m∠YWX
11 m∠ZYW = m∠YWX if m∠ZYW = m∠PWX and m∠PWX = m∠YWX, then m∠ZYW = m∠YWX
12 ZY || WX if m∠ZYW = m∠YWX, then ZY || WX
13 WX || ZY if ZY || WX, then WX || ZY
14 WXYZ is a parallelogram if WX || ZY and distance WX = distance YZ, then WXYZ is a parallelogram

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