Proof: If Angles Congruent Then Parallelogram

Let's prove the following theorem:

Proof:

Proof Table

#	Claim	Reason
1	quadrilateral WXYZ is convex	if m∠WXY = m∠YZW and m∠XYZ = m∠ZWX, then quadrilateral WXYZ is convex (If Opposite Angles Are Equal Then Quad is Convex)
2	(((m∠WXY) + (m∠XYZ)) + (m∠YZW)) + (m∠ZWX) = 360	if quadrilateral WXYZ is convex, then (((m∠WXY) + (m∠XYZ)) + (m∠YZW)) + (m∠ZWX) = 360 (Sum of Angles in Quadrilateral is 360)
3	(((m∠WXY) + (m∠XYZ)) + (m∠YZW)) + (m∠XYZ) = 360	if (((m∠WXY) + (m∠XYZ)) + (m∠YZW)) + (m∠ZWX) = 360 and m∠XYZ = m∠ZWX, then (((m∠WXY) + (m∠XYZ)) + (m∠YZW)) + (m∠XYZ) = 360 (Substitute 2)
4	((m∠WXY) + (m∠XYZ)) + (m∠YZW) = ((m∠WXY) + (m∠XYZ)) + (m∠WXY)	if m∠WXY = m∠YZW, then ((m∠WXY) + (m∠XYZ)) + (m∠YZW) = ((m∠WXY) + (m∠XYZ)) + (m∠WXY) (Add Term to Both Sides 2)
5	(((m∠WXY) + (m∠XYZ)) + (m∠YZW)) + (m∠XYZ) = (((m∠WXY) + (m∠XYZ)) + (m∠WXY)) + (m∠XYZ)	if ((m∠WXY) + (m∠XYZ)) + (m∠YZW) = ((m∠WXY) + (m∠XYZ)) + (m∠WXY), then (((m∠WXY) + (m∠XYZ)) + (m∠YZW)) + (m∠XYZ) = (((m∠WXY) + (m∠XYZ)) + (m∠WXY)) + (m∠XYZ) (Additive Property of Equality)
6	(((m∠WXY) + (m∠XYZ)) + (m∠WXY)) + (m∠XYZ) = 360	if (((m∠WXY) + (m∠XYZ)) + (m∠YZW)) + (m∠XYZ) = (((m∠WXY) + (m∠XYZ)) + (m∠WXY)) + (m∠XYZ) and (((m∠WXY) + (m∠XYZ)) + (m∠YZW)) + (m∠XYZ) = 360, then (((m∠WXY) + (m∠XYZ)) + (m∠WXY)) + (m∠XYZ) = 360 (Transitive Property of Equality Variation 2)
7	(m∠WXY) + (m∠XYZ) = 180	if (((m∠WXY) + (m∠XYZ)) + (m∠WXY)) + (m∠XYZ) = 360, then (m∠WXY) + (m∠XYZ) = 180 (Reorder Terms 6)
8	(m∠WXY) + (m∠ZYX) = 180	if (m∠WXY) + (m∠XYZ) = 180, then (m∠WXY) + (m∠ZYX) = 180 (Reorder Terms 7)
9	∠WXY and ∠ZYX are supplementary	if (m∠WXY) + (m∠ZYX) = 180, then ∠WXY and ∠ZYX are supplementary (Supplementary Angles Property (Converse))
10	WX || ZY	if ∠WXY and ∠ZYX are supplementary, then WX || ZY (Consecutive Interior Angles Theorem)
11	(m∠WXY) + (m∠ZWX) = 180	if (m∠WXY) + (m∠XYZ) = 180 and m∠XYZ = m∠ZWX, then (m∠WXY) + (m∠ZWX) = 180 (Substitution Example 10)
12	(m∠ZWX) + (m∠WXY) = 180	if (m∠WXY) + (m∠ZWX) = 180, then (m∠ZWX) + (m∠WXY) = 180 (Commutative Property Example 2)
13	m∠WXY = m∠YXW	m∠WXY = m∠YXW (Angle Symmetry Property)
14	(m∠ZWX) + (m∠YXW) = 180	if (m∠ZWX) + (m∠WXY) = 180 and m∠WXY = m∠YXW, then (m∠ZWX) + (m∠YXW) = 180 (Substitution Example 10)
15	∠ZWX and ∠YXW are supplementary	if (m∠ZWX) + (m∠YXW) = 180, then ∠ZWX and ∠YXW are supplementary (Supplementary Angles Property (Converse))
16	ZW || YX	if ∠ZWX and ∠YXW are supplementary, then ZW || YX (Consecutive Interior Angles Theorem)
17	WZ || XY	if ZW || YX, then WZ || XY (Parallel Lines Property 2)
18	WXYZ is a parallelogram	if WX || ZY and WZ || XY, then WXYZ is a parallelogram (Parallelogram Property)

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