Proof: Hypothenuse-Leg (HL) Theorem

Let's prove the following theorem:

if ∠ABC is a right angle and ∠XYZ is a right angle and distance AC = distance XZ and distance BC = distance YZ, then △ABC ≅ △XYZ

B A C X Y Z P

Proof:

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Given
1 ∠ABC is a right angle
2 ∠XYZ is a right angle
3 distance AC = distance XZ
4 distance BC = distance YZ
Additional Assumptions
5 distance AB = distance PY
6 m∠XYP = 180
Proof Table
# Claim Reason
1 m∠ABC = 90 if ∠ABC is a right angle, then m∠ABC = 90
2 m∠XYZ = 90 if ∠XYZ is a right angle, then m∠XYZ = 90
3 ∠XYZ and ∠ZYP are supplementary if m∠XYP = 180, then ∠XYZ and ∠ZYP are supplementary
4 (m∠XYZ) + (m∠ZYP) = 180 if ∠XYZ and ∠ZYP are supplementary, then (m∠XYZ) + (m∠ZYP) = 180
5 90 + (m∠ZYP) = 180 if (m∠XYZ) + (m∠ZYP) = 180 and m∠XYZ = 90, then 90 + (m∠ZYP) = 180
6 m∠ZYP = 90 if 90 + (m∠ZYP) = 180, then m∠ZYP = 90
7 m∠PYZ = 90 if m∠ZYP = 90, then m∠PYZ = 90
8 m∠ABC = m∠PYZ if m∠ABC = 90 and m∠PYZ = 90, then m∠ABC = m∠PYZ
9 △ABC ≅ △PYZ if distance AB = distance PY and m∠ABC = m∠PYZ and distance BC = distance YZ, then △ABC ≅ △PYZ
10 m∠XYZ = m∠PYZ if m∠XYZ = 90 and m∠PYZ = 90, then m∠XYZ = m∠PYZ
11 distance AC = distance PZ if △ABC ≅ △PYZ, then distance AC = distance PZ
12 distance XZ = distance PZ if distance AC = distance XZ and distance AC = distance PZ, then distance XZ = distance PZ
13 m∠ZXP = m∠ZPX if distance XZ = distance PZ, then m∠ZXP = m∠ZPX
14 m∠ZXY = m∠ZXP if m∠XYP = 180, then m∠ZXY = m∠ZXP
15 m∠ZPY = m∠ZPX if m∠XYP = 180, then m∠ZPY = m∠ZPX
16 m∠ZXY = m∠ZPY if m∠ZXY = m∠ZXP and m∠ZPY = m∠ZPX and m∠ZXP = m∠ZPX, then m∠ZXY = m∠ZPY
17 △YXZ ≅ △YPZ if m∠XYZ = m∠PYZ and m∠ZXY = m∠ZPY and distance XZ = distance PZ, then △YXZ ≅ △YPZ
18 △XYZ ≅ △PYZ if △YXZ ≅ △YPZ, then △XYZ ≅ △PYZ
19 △ABC ≅ △XYZ if △ABC ≅ △PYZ and △XYZ ≅ △PYZ, then △ABC ≅ △XYZ

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