Proof: Triangles Sum to 180

Let's prove the following theorem:

Proof:

Proof Table

#	Claim	Reason
1	m∠ZXW = m∠XWY	if ZX || WY, then m∠ZXW = m∠XWY (Alternate Interior Angles Theorem (Converse) 2)
2	XP || WY	if ZX || WY and m∠ZXP = 180, then XP || WY (Parallel Lines Property 4)
3	PX || YW	if XP || WY, then PX || YW (Parallel Lines Property 2)
4	m∠PXY = m∠XYW	if PX || YW, then m∠PXY = m∠XYW (Alternate Interior Angles Theorem (Converse) 2)
5	point W lies in interior of ∠ZXY	if ZX || WY, then point W lies in interior of ∠ZXY (A Parallel Point That is in the Interior of an Angle)
6	m∠ZXY = (m∠ZXW) + (m∠WXY)	if point W lies in interior of ∠ZXY, then m∠ZXY = (m∠ZXW) + (m∠WXY) (Angle Addition Theorem)
7	m∠ZXP = (m∠ZXY) + (m∠YXP)	if m∠ZXP = 180, then m∠ZXP = (m∠ZXY) + (m∠YXP) (Collinear Points Property 2)
8	(m∠ZXY) + (m∠YXP) = 180	if m∠ZXP = (m∠ZXY) + (m∠YXP) and m∠ZXP = 180, then (m∠ZXY) + (m∠YXP) = 180 (Transitive Property of Equality Variation 2)
9	((m∠ZXW) + (m∠WXY)) + (m∠YXP) = 180	if (m∠ZXY) + (m∠YXP) = 180 and m∠ZXY = (m∠ZXW) + (m∠WXY), then ((m∠ZXW) + (m∠WXY)) + (m∠YXP) = 180 (Substitution 8)
10	m∠YXP = m∠XYW	if m∠PXY = m∠XYW, then m∠YXP = m∠XYW (Angle Symmetry B)
11	((m∠ZXW) + (m∠WXY)) + (m∠XYW) = 180	if ((m∠ZXW) + (m∠WXY)) + (m∠YXP) = 180 and m∠YXP = m∠XYW, then ((m∠ZXW) + (m∠WXY)) + (m∠XYW) = 180 (Substitution Example 10)
12	(m∠ZXW) + (m∠WXY) = (m∠XWY) + (m∠WXY)	if m∠ZXW = m∠XWY, then (m∠ZXW) + (m∠WXY) = (m∠XWY) + (m∠WXY) (Additive Property of Equality)
13	((m∠XWY) + (m∠WXY)) + (m∠XYW) = 180	if ((m∠ZXW) + (m∠WXY)) + (m∠XYW) = 180 and (m∠ZXW) + (m∠WXY) = (m∠XWY) + (m∠WXY), then ((m∠XWY) + (m∠WXY)) + (m∠XYW) = 180 (Substitution 8)
14	m∠XWY = m∠YWX	m∠XWY = m∠YWX (Angle Symmetry Property)
15	((m∠YWX) + (m∠WXY)) + (m∠XYW) = 180	if ((m∠XWY) + (m∠WXY)) + (m∠XYW) = 180 and m∠XWY = m∠YWX, then ((m∠YWX) + (m∠WXY)) + (m∠XYW) = 180 (Substitute First Term)

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