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

W X Y Z S T

Proof:

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Given
1 quadrilateral WXYZ is a trapezoid
2 distance WY = distance XZ
Additional Assumptions
3 m∠ZST = 90
4 m∠YTS = 90
5 m∠WST = 180
6 m∠STX = 180
Proof Table
# 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

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