Proof: If Equiangular Then Rectangle
Let's prove the following theorem:
if m∠WXY = m∠XYZ and m∠XYZ = m∠YZW and m∠YZW = m∠ZWX, then WXYZ is a rectangle
Proof:
Proof Table
| # | Claim | Reason |
|---|---|---|
| 1 | m∠WXY = m∠YZW | if m∠WXY = m∠XYZ and m∠XYZ = m∠YZW, then m∠WXY = m∠YZW |
| 2 | m∠XYZ = m∠ZWX | if m∠XYZ = m∠YZW and m∠YZW = m∠ZWX, then m∠XYZ = m∠ZWX |
| 3 | WXYZ is a parallelogram | if m∠XYZ = m∠ZWX and m∠WXY = m∠YZW, then WXYZ is a parallelogram |
| 4 | quadrilateral WXYZ is convex | if m∠WXY = m∠YZW and m∠XYZ = m∠ZWX, then quadrilateral WXYZ is convex |
| 5 | (((m∠WXY) + (m∠XYZ)) + (m∠YZW)) + (m∠ZWX) = 360 | if quadrilateral WXYZ is convex, then (((m∠WXY) + (m∠XYZ)) + (m∠YZW)) + (m∠ZWX) = 360 |
| 6 | m∠WXY = m∠ZWX | if m∠WXY = m∠XYZ and m∠XYZ = m∠ZWX, then m∠WXY = m∠ZWX |
| 7 | (((m∠WXY) + (m∠XYZ)) + (m∠YZW)) + (m∠WXY) = 360 | if (((m∠WXY) + (m∠XYZ)) + (m∠YZW)) + (m∠ZWX) = 360 and m∠WXY = m∠ZWX, then (((m∠WXY) + (m∠XYZ)) + (m∠YZW)) + (m∠WXY) = 360 |
| 8 | m∠YZW = m∠WXY | if m∠WXY = m∠YZW, then m∠YZW = m∠WXY |
| 9 | (((m∠WXY) + (m∠XYZ)) + (m∠WXY)) + (m∠WXY) = 360 | if m∠YZW = m∠WXY and (((m∠WXY) + (m∠XYZ)) + (m∠YZW)) + (m∠WXY) = 360, then (((m∠WXY) + (m∠XYZ)) + (m∠WXY)) + (m∠WXY) = 360 |
| 10 | (m∠WXY) + (m∠XYZ) = (m∠WXY) + (m∠WXY) | if m∠WXY = m∠XYZ, then (m∠WXY) + (m∠XYZ) = (m∠WXY) + (m∠WXY) |
| 11 | (((m∠WXY) + (m∠WXY)) + (m∠WXY)) + (m∠WXY) = 360 | if (((m∠WXY) + (m∠XYZ)) + (m∠WXY)) + (m∠WXY) = 360 and (m∠WXY) + (m∠XYZ) = (m∠WXY) + (m∠WXY), then (((m∠WXY) + (m∠WXY)) + (m∠WXY)) + (m∠WXY) = 360 |
| 12 | m∠WXY = 90 | if (((m∠WXY) + (m∠WXY)) + (m∠WXY)) + (m∠WXY) = 360, then m∠WXY = 90 |
| 13 | ∠WXY is a right angle | if m∠WXY = 90, then ∠WXY is a right angle |
| 14 | WXYZ is a rectangle | if WXYZ is a parallelogram and ∠WXY is a right angle, then WXYZ is a rectangle |
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