Proof: SAS Sides

Let's prove the following theorem:

if m∠ABC = 180 and m∠DCB = 180 and ∠XAC is a right angle and ∠YDB is a right angle and distance XA = distance YD and distance AB = distance DC, then distance XC = distance YB

Proof:

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Given

1	m∠ABC = 180
2	m∠DCB = 180
3	∠XAC is a right angle
4	∠YDB is a right angle
5	distance XA = distance YD
6	distance AB = distance DC

Proof Table

#	Claim	Reason
1	distance BC = distance CB	distance BC = distance CB (Distance Equality Property)
2	distance AC = (distance AB) + (distance BC)	if m∠ABC = 180, then distance AC = (distance AB) + (distance BC) (Collinear Points Property)
3	distance AC = (distance DC) + (distance BC)	if distance AC = (distance AB) + (distance BC) and distance AB = distance DC, then distance AC = (distance DC) + (distance BC) (Term 1 Substitution)
4	distance AC = (distance DC) + (distance CB)	if distance AC = (distance DC) + (distance BC) and distance BC = distance CB, then distance AC = (distance DC) + (distance CB) (Term 2 Substitution)
5	distance DB = (distance DC) + (distance CB)	if m∠DCB = 180, then distance DB = (distance DC) + (distance CB) (Collinear Points Property)
6	distance AC = distance DB	if distance AC = (distance DC) + (distance CB) and distance DB = (distance DC) + (distance CB), then distance AC = distance DB (Transitive Property of Equality Variation 1)
7	m∠XAC = m∠YDB	if ∠XAC is a right angle and ∠YDB is a right angle, then m∠XAC = m∠YDB (Right Angle Theorem)
8	△XAC ≅ △YDB	if distance XA = distance YD and m∠XAC = m∠YDB and distance AC = distance DB, then △XAC ≅ △YDB (SAS Property)
9	distance XC = distance YB	if △XAC ≅ △YDB, then distance XC = distance YB (Congruent Triangles to Distance 3)

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