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
Proof:
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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