Proof: Bisector Angle

Let's prove the following theorem:

if PX ⊥ XY and PZ ⊥ ZY and distance XP = distance ZP, then m∠PYX = m∠PYZ

Y X P Z

Proof:

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Given
1 PX ⊥ XY
2 PZ ⊥ ZY
3 distance XP = distance ZP
Proof Table
# Claim Reason
1 ∠PZY is a right angle if PZ ⊥ ZY, then ∠PZY is a right angle
2 ∠YZP is a right angle if ∠PZY is a right angle, then ∠YZP is a right angle
3 ∠PXY is a right angle if PX ⊥ XY, then ∠PXY is a right angle
4 ∠YXP is a right angle if ∠PXY is a right angle, then ∠YXP is a right angle
5 distance YP = distance YP distance YP = distance YP
6 △YXP ≅ △YZP if ∠YXP is a right angle and ∠YZP is a right angle and distance YP = distance YP and distance XP = distance ZP, then △YXP ≅ △YZP
7 m∠PYX = m∠PYZ if △YXP ≅ △YZP, then m∠PYX = m∠PYZ

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