Proof: Triangle Proportionality Theorem (Converse)

Let's prove the following theorem:

if (distance ZS) / (distance ZX) = (distance ZT) / (distance ZY) and m∠XSZ = 180 and m∠YTZ = 180, then ST || XY

X Y Z S T

Proof:

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Given
1 (distance ZS) / (distance ZX) = (distance ZT) / (distance ZY)
2 m∠XSZ = 180
3 m∠YTZ = 180
Proof Table
# Claim Reason
1 (distance ZT) / (distance ZY) = (distance TZ) / (distance YZ) (distance ZT) / (distance ZY) = (distance TZ) / (distance YZ)
2 (distance TZ) / (distance YZ) = (distance ZS) / (distance ZX) if (distance ZS) / (distance ZX) = (distance ZT) / (distance ZY) and (distance ZT) / (distance ZY) = (distance TZ) / (distance YZ), then (distance TZ) / (distance YZ) = (distance ZS) / (distance ZX)
3 m∠YZX = m∠TZS if m∠YTZ = 180 and m∠XSZ = 180, then m∠YZX = m∠TZS
4 m∠TZS = m∠YZX if m∠YZX = m∠TZS, then m∠TZS = m∠YZX
5 △TZS ∼ △YZX if m∠TZS = m∠YZX and (distance TZ) / (distance YZ) = (distance ZS) / (distance ZX), then △TZS ∼ △YZX
6 m∠ZST = m∠ZXY if △TZS ∼ △YZX, then m∠ZST = m∠ZXY
7 ST || XY if m∠ZST = m∠ZXY, then ST || XY

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