Proof: If Diagonals Congreuent Then Isosceles Trapezoid
Let's prove the following theorem:
if quadrilateral WXYZ is a trapezoid and distance WY = distance XZ, then distance ZW = distance YX
Proof:
Given
Assumptions
| 1 | quadrilateral WXYZ is a trapezoid |
|---|---|
| 2 | distance WY = distance XZ |
| 3 | m∠ZST = 90 |
|---|---|
| 4 | m∠YTS = 90 |
| 5 | m∠WST = 180 |
| 6 | m∠STX = 180 |
| # | Claim | Reason |
|---|---|---|
| 1 | (m∠ZST) + (m∠YTS) = 180 | if m∠ZST = 90 and m∠YTS = 90, then (m∠ZST) + (m∠YTS) = 180 |
| 2 | ∠ZST and ∠YTS are supplementary | if (m∠ZST) + (m∠YTS) = 180, then ∠ZST and ∠YTS are supplementary |
| 3 | ZS || YT | if ∠ZST and ∠YTS are supplementary, then ZS || YT |
| 4 | WX || ZY | if quadrilateral WXYZ is a trapezoid, then WX || ZY |
| 5 | ST || ZY | if WX || ZY and m∠WST = 180 and m∠STX = 180, then ST || ZY |
| 6 | ZY || ST | if ST || ZY, then ZY || ST |
| 7 | ZSTY is a parallelogram | if ZS || YT and ZY || ST, then ZSTY is a parallelogram |
| 8 | distance ZS = distance YT | if ZSTY is a parallelogram, then distance ZS = distance YT |
| 9 | distance SZ = distance TY | if distance ZS = distance YT, then distance SZ = distance TY |
| 10 | m∠ZST = m∠ZSX | if m∠STX = 180, then m∠ZST = m∠ZSX |
| 11 | m∠XSZ = 90 | if m∠ZST = 90 and m∠ZST = m∠ZSX, then m∠XSZ = 90 |
| 12 | ∠XSZ is a right angle | if m∠XSZ = 90, then ∠XSZ is a right angle |
| 13 | m∠YTS = m∠YTW | if m∠WST = 180, then m∠YTS = m∠YTW |
| 14 | m∠WTY = 90 | if m∠YTS = 90 and m∠YTS = m∠YTW, then m∠WTY = 90 |
| 15 | ∠WTY is a right angle | if m∠WTY = 90, then ∠WTY is a right angle |
| 16 | distance XZ = distance WY | if distance WY = distance XZ, then distance XZ = distance WY |
| 17 | △XSZ ≅ △WTY | if ∠XSZ is a right angle and ∠WTY is a right angle and distance XZ = distance WY and distance SZ = distance TY, then △XSZ ≅ △WTY |
| 18 | m∠ZXS = m∠YWT | if △XSZ ≅ △WTY, then m∠ZXS = m∠YWT |
| 19 | m∠WSX = 180 | if m∠WST = 180 and m∠STX = 180, then m∠WSX = 180 |
| 20 | m∠WTX = 180 | if m∠WST = 180 and m∠STX = 180, then m∠WTX = 180 |
| 21 | m∠ZXW = m∠YWX | if m∠ZXS = m∠YWT and m∠WSX = 180 and m∠WTX = 180, then m∠ZXW = m∠YWX |
| 22 | distance XW = distance WX | distance XW = distance WX |
| 23 | distance ZX = distance YW | if △XSZ ≅ △WTY, then distance ZX = distance YW |
| 24 | △ZXW ≅ △YWX | if distance ZX = distance YW and m∠ZXW = m∠YWX and distance XW = distance WX, then △ZXW ≅ △YWX |
| 25 | distance ZW = distance YX | if △ZXW ≅ △YWX, then distance ZW = distance YX |
Comments
Please log in to add comments