Proof: Bisector Point Equidistant From Sides

Let's prove the following theorem:

if m∠PXY = 90 and m∠PZY = 90 and ray YP bisects ∠XYZ, then distance PX = distance PZ

P Z Y X

Proof:

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Given
1 m∠PXY = 90
2 m∠PZY = 90
3 ray YP bisects ∠XYZ
Proof Table
# Claim Reason
1 m∠XYP = m∠PYZ if ray YP bisects ∠XYZ, then m∠XYP = m∠PYZ
2 m∠XYP = m∠ZYP if m∠XYP = m∠PYZ, then m∠XYP = m∠ZYP
3 m∠PXY = m∠PZY if m∠PXY = 90 and m∠PZY = 90, then m∠PXY = m∠PZY
4 distance YP = distance YP distance YP = distance YP
5 △XYP ≅ △ZYP if m∠PXY = m∠PZY and m∠XYP = m∠ZYP and distance YP = distance YP, then △XYP ≅ △ZYP
6 distance PX = distance PZ if △XYP ≅ △ZYP, then distance PX = distance PZ

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