Proof: Altitudes Isosceles

Let's prove the following theorem:

if distance YX = distance YZ and XT ⊥ TZ and ZS ⊥ SX and m∠XSY = 180 and m∠ZTY = 180, then distance ZS = distance XT

X S Y T Z

Proof:

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Given
1 distance YX = distance YZ
2 XT ⊥ TZ
3 ZS ⊥ SX
4 m∠XSY = 180
5 m∠ZTY = 180
Proof Table
# Claim Reason
1 m∠YXZ = m∠YZX if distance YX = distance YZ, then m∠YXZ = m∠YZX
2 m∠SXZ = m∠YXZ if m∠XSY = 180, then m∠SXZ = m∠YXZ
3 m∠TZX = m∠YZX if m∠ZTY = 180, then m∠TZX = m∠YZX
4 m∠SXZ = m∠YZX if m∠SXZ = m∠YXZ and m∠YXZ = m∠YZX, then m∠SXZ = m∠YZX
5 m∠SXZ = m∠TZX if m∠SXZ = m∠YZX and m∠TZX = m∠YZX, then m∠SXZ = m∠TZX
6 m∠ZSX = 90 if ZS ⊥ SX, then m∠ZSX = 90
7 m∠XTZ = 90 if XT ⊥ TZ, then m∠XTZ = 90
8 m∠ZSX = m∠XTZ if m∠ZSX = 90 and m∠XTZ = 90, then m∠ZSX = m∠XTZ
9 distance XZ = distance ZX distance XZ = distance ZX
10 △SXZ ≅ △TZX if m∠ZSX = m∠XTZ and m∠SXZ = m∠TZX and distance XZ = distance ZX, then △SXZ ≅ △TZX
11 distance ZS = distance XT if △SXZ ≅ △TZX, then distance ZS = distance XT

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