Proof: If Bisector Then Parallelogram

Let's prove the following theorem:

Proof:

Proof Table

#	Claim	Reason
1	m∠WPX = m∠YPZ	if m∠WPY = 180 and m∠XPZ = 180, then m∠WPX = m∠YPZ (Vertical Angles C)
2	distance WP = distance YP	if distance WP = distance PY, then distance WP = distance YP (Distance Property 2)
3	distance PX = distance PZ	if distance XP = distance PZ, then distance PX = distance PZ (Distance Property 1)
4	△WPX ≅ △YPZ	if distance WP = distance YP and m∠WPX = m∠YPZ and distance PX = distance PZ, then △WPX ≅ △YPZ (SAS Property)
5	distance WX = distance YZ	if △WPX ≅ △YPZ, then distance WX = distance YZ (Congruent Triangles to Distance 3)
6	m∠XWP = m∠ZYP	if △WPX ≅ △YPZ, then m∠XWP = m∠ZYP (Congruent Triangles Property 5)
7	m∠ZYP = m∠PWX	if m∠XWP = m∠ZYP, then m∠ZYP = m∠PWX (Angle Symmetry 4)
8	m∠ZYP = m∠ZYW	if m∠WPY = 180, then m∠ZYP = m∠ZYW (Collinear Angles Property 10)
9	m∠ZYW = m∠PWX	if m∠ZYP = m∠ZYW and m∠ZYP = m∠PWX, then m∠ZYW = m∠PWX (Transitive Property of Equality Variation 2)
10	m∠PWX = m∠YWX	if m∠WPY = 180, then m∠PWX = m∠YWX (Collinear Angles Property 9)
11	m∠ZYW = m∠YWX	if m∠ZYW = m∠PWX and m∠PWX = m∠YWX, then m∠ZYW = m∠YWX (Transitive Property of Equality)
12	ZY || WX	if m∠ZYW = m∠YWX, then ZY || WX (Alternate Interior Angles Theorem 4)
13	WX || ZY	if ZY || WX, then WX || ZY (Parallel Lines Property 3)
14	WXYZ is a parallelogram	if WX || ZY and distance WX = distance YZ, then WXYZ is a parallelogram (If Congruent And Parallel Then Parallelogram)

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