Proof: If Sas Then Similar Triangles

Let's prove the following theorem:

if m∠ABC = m∠XYZ and (distance AB) / (distance XY) = (distance BC) / (distance YZ), then △ABC ∼ △XYZ

C B A Z X Y S T

Proof:

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Given
1 m∠ABC = m∠XYZ
2 (distance AB) / (distance XY) = (distance BC) / (distance YZ)
Assumptions
3 distance BC = distance YS
4 ST || ZX
5 m∠ZSY = 180
6 m∠XTY = 180
Proof Table
# Claim Reason
1 m∠ZXY = m∠STY if ST || ZX and m∠XTY = 180, then m∠ZXY = m∠STY
2 m∠XZY = m∠TSY if ST || ZX and m∠ZSY = 180, then m∠XZY = m∠TSY
3 m∠YZX = m∠YST if m∠XZY = m∠TSY, then m∠YZX = m∠YST
4 △ZXY ∼ △STY if m∠YZX = m∠YST and m∠ZXY = m∠STY, then △ZXY ∼ △STY
5 △XYZ ∼ △TYS if △ZXY ∼ △STY, then △XYZ ∼ △TYS
6 △TYS ∼ △XYZ if △XYZ ∼ △TYS, then △TYS ∼ △XYZ
7 (distance YS) / (distance YZ) = (distance TY) / (distance XY) if △ZXY ∼ △STY, then (distance YS) / (distance YZ) = (distance TY) / (distance XY)
8 (distance AB) / (distance XY) = (distance YS) / (distance YZ) if (distance AB) / (distance XY) = (distance BC) / (distance YZ) and distance BC = distance YS, then (distance AB) / (distance XY) = (distance YS) / (distance YZ)
9 (distance AB) / (distance XY) = (distance TY) / (distance XY) if (distance AB) / (distance XY) = (distance YS) / (distance YZ) and (distance YS) / (distance YZ) = (distance TY) / (distance XY), then (distance AB) / (distance XY) = (distance TY) / (distance XY)
10 distance TY = distance AB if (distance AB) / (distance XY) = (distance TY) / (distance XY), then distance TY = distance AB
11 distance AB = distance TY if distance TY = distance AB, then distance AB = distance TY
12 m∠XYZ = m∠TYS if △XYZ ∼ △TYS, then m∠XYZ = m∠TYS
13 m∠ABC = m∠TYS if m∠ABC = m∠XYZ and m∠XYZ = m∠TYS, then m∠ABC = m∠TYS
14 △ABC ≅ △TYS if distance AB = distance TY and m∠ABC = m∠TYS and distance BC = distance YS, then △ABC ≅ △TYS
15 △ABC ∼ △TYS if △ABC ≅ △TYS, then △ABC ∼ △TYS
16 △ABC ∼ △XYZ if △ABC ∼ △TYS and △TYS ∼ △XYZ, then △ABC ∼ △XYZ

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