Proof: Perpendicular Bisector Theorem

Let's prove the following theorem:

Proof:

Proof Table

#	Claim	Reason
1	m∠XMY = 180	if M is the midpoint of line XY, then m∠XMY = 180 (Angle Formed by Midpoint)
2	∠SMY is a right angle	if SM ⊥ MY, then ∠SMY is a right angle (Perpendicular Angles)
3	m∠SMY = 90	if ∠SMY is a right angle, then m∠SMY = 90 (Right Angle Property)
4	∠XMS and ∠SMY are supplementary	if m∠XMY = 180, then ∠XMS and ∠SMY are supplementary (Supplementary Angles Theorem (Converse))
5	(m∠XMS) + (m∠SMY) = 180	if ∠XMS and ∠SMY are supplementary, then (m∠XMS) + (m∠SMY) = 180 (Supplementary Angles Property)
6	(m∠XMS) + 90 = 180	if (m∠XMS) + (m∠SMY) = 180 and m∠SMY = 90, then (m∠XMS) + 90 = 180 (Substitution Example 10)
7	m∠XMS = 90	if (m∠XMS) + 90 = 180, then m∠XMS = 90 (Add Number to Both Sides)
8	m∠SMY = m∠XMS	if m∠SMY = 90 and m∠XMS = 90, then m∠SMY = m∠XMS (Transitive Property of Equality Variation 1)
9	m∠XMS = m∠YMS	if m∠SMY = m∠XMS, then m∠XMS = m∠YMS (Angle Symmetry 4)
10	distance XM = distance MY	if M is the midpoint of line XY, then distance XM = distance MY (Midpoint Property)
11	distance XM = distance YM	if distance XM = distance MY, then distance XM = distance YM (Distance Property 2)
12	distance MS = distance MS	distance MS = distance MS (Reflexive Property of Equality)
13	△XMS ≅ △YMS	if distance XM = distance YM and m∠XMS = m∠YMS and distance MS = distance MS, then △XMS ≅ △YMS (SAS Property)
14	distance SX = distance SY	if △XMS ≅ △YMS, then distance SX = distance SY (Congruent Triangles Property 9)

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