Proof: Perpendicular Bisector Theorem (Converse)
Let's prove the following theorem:
if distance XS = distance YS and M is the midpoint of line XY, then m∠XMS = 90
Proof:
Proof Table
| # | Claim | Reason |
|---|---|---|
| 1 | distance SM = distance SM | distance SM = distance SM |
| 2 | distance XM = distance MY | if M is the midpoint of line XY, then distance XM = distance MY |
| 3 | distance MX = distance MY | if distance XM = distance MY, then distance MX = distance MY |
| 4 | △XSM ≅ △YSM | if distance XS = distance YS and distance SM = distance SM and distance MX = distance MY, then △XSM ≅ △YSM |
| 5 | m∠SMX = m∠SMY | if △XSM ≅ △YSM, then m∠SMX = m∠SMY |
| 6 | m∠XMS = m∠SMY | if m∠SMX = m∠SMY, then m∠XMS = m∠SMY |
| 7 | m∠XMY = 180 | if M is the midpoint of line XY, then m∠XMY = 180 |
| 8 | ∠XMS and ∠SMY are supplementary | if m∠XMY = 180, then ∠XMS and ∠SMY are supplementary |
| 9 | (m∠XMS) + (m∠SMY) = 180 | if ∠XMS and ∠SMY are supplementary, then (m∠XMS) + (m∠SMY) = 180 |
| 10 | (m∠XMS) + (m∠XMS) = 180 | if (m∠XMS) + (m∠SMY) = 180 and m∠XMS = m∠SMY, then (m∠XMS) + (m∠XMS) = 180 |
| 11 | m∠XMS = 90 | if (m∠XMS) + (m∠XMS) = 180, then m∠XMS = 90 |
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