Proofs
A proof is a series of claims that lead to a conclusion. Some proofs are conditional, which means that the claims can only be made under certain conditions. Click on a statement to see the proof
if (distance AB) / (distance XY) = (distance BC) / (distance YZ) and (distance BC) / (distance YZ) = (distance CA) / (distance ZX) and distance AC = distance SZ and ST || XY and m∠XSZ = 180 and m∠YTZ = 180, then △ABC ∼ △XYZ
if m∠ABC = m∠XYZ and (distance AB) / (distance XY) = (distance BC) / (distance YZ), then △ABC ∼ △XYZ
if (distance ZS) / (distance ZX) = (distance ZT) / (distance ZY) and m∠XSZ = 180 and m∠YTZ = 180, then ST || XY
if △ABC ∼ △XYZ and ∠BPC is a right angle and ∠YQZ is a right angle and m∠APC = 180 and m∠XQZ = 180, then (distance BP) / (distance YQ) = (distance BC) / (distance YZ)
if △ABC ∼ △XYZ and M is the midpoint of line AC and N is the midpoint of line XZ, then (distance BM) / (distance YN) = (distance BC) / (distance YZ)
if △ABC ∼ △XYZ and ray BP bisects ∠ABC and ray YQ bisects ∠XYZ and m∠APC = 180 and m∠XQZ = 180, then (distance BC) / (distance YZ) = (distance BP) / (distance YQ)
if ∠ZXY is a right angle and ∠XPY is a right angle and m∠YPZ = 180, then △PYX ∼ △XYZ
if ∠ZXY is a right angle and ∠XPY is a right angle and m∠YPZ = 180, then △PXZ ∼ △XYZ
if ∠ZXY is a right angle, then ((distance XY) ⋅ (distance XY)) + ((distance XZ) ⋅ (distance XZ)) = (distance YZ) ⋅ (distance YZ)
if ∠CAB is a right angle, then (distance BC) ⋅ (distance BC) = ((distance AB) ⋅ (distance AB)) + ((distance AC) ⋅ (distance AC))
if ∠CAB is a right angle, then (distance CB) ⋅ (distance CB) = ((distance CA) ⋅ (distance CA)) + ((distance AB) ⋅ (distance AB))
if ∠ABC is a right angle, then (distance AC) ⋅ (distance AC) = ((distance AB) ⋅ (distance AB)) + ((distance BC) ⋅ (distance BC))
if m∠WSX = 180 and m∠YTZ = 180 and m∠WST = m∠STZ, then WX || YZ
if m∠WSX = 180 and m∠YTZ = 180 and m∠WST = m∠STZ, then XW || ZY
if m∠WSX = 180 and m∠YTZ = 180 and m∠YTS = m∠TSX, then WX || YZ
if m∠WSX = 180 and m∠YTZ = 180 and m∠STZ = m∠TSW, then WX || YZ
if m∠WST = m∠STZ, then WS || TZ
if m∠WJI = m∠YKJ and m∠WJX = 180 and m∠YKZ = 180 and m∠IJK = 180, then WX || YZ
if m∠WJI = m∠YKJ and m∠WJX = 180 and m∠YKZ = 180 and m∠KJI = 180, then WX || YZ
if m∠YKJ = m∠WJI and m∠WJX = 180 and m∠YKZ = 180 and m∠KJI = 180, then YZ || WX
if ∠WST and ∠YTS are supplementary, then WS || YT
if WX ⊥ XS and YZ ⊥ ZT and m∠WXG = 180 and m∠YZH = 180 and m∠SXZ = 180 and m∠XZT = 180, then YH || WG
if WX || YZ and m∠WSX = 180 and m∠YTZ = 180, then m∠WST = m∠STZ
if WX || YZ and m∠WSX = 180 and m∠YTZ = 180, then m∠XST = m∠STY
if WS || TZ, then m∠WST = m∠STZ
if WS || TZ, then m∠WST = m∠ZTS
if WS || TZ, then m∠TSW = m∠STZ
if WX || YZ, then m∠ZYX = m∠YXW
if WS || TZ, then m∠SWZ = m∠WZT
if WS || TZ, then m∠TZW = m∠SWZ
if WS || TZ, then m∠SWZ = m∠TZW
if WX || YZ, then m∠YZW = m∠ZWX
if WX || YZ and m∠WSX = 180 and m∠YTZ = 180 and m∠RST = 180, then m∠WSR = m∠STY
if WX || YZ and m∠WSX = 180 and m∠YTZ = 180 and m∠RST = 180, then m∠WSR = m∠YTS
if WX || YZ and m∠RXZ = 180, then m∠WXR = m∠YZX
if WX || YZ and m∠WYR = 180, then m∠ZYR = m∠XWY
if WX || YZ and m∠ZXR = 180, then m∠YZX = m∠WXR
if WX || YZ and m∠ZXR = 180, then m∠YZR = m∠WXR
if WX || YZ and m∠YWR = 180, then m∠ZYW = m∠XWR
if WX || YZ and m∠YWR = 180, then m∠ZYR = m∠XWR
if WX || YZ and m∠WSX = 180 and m∠YTZ = 180, then ∠WST and ∠STY are supplementary
if WX || YZ, then ∠WXZ and ∠XZY are supplementary
if WX || YZ and m∠WSX = 180 and m∠YTZ = 180 and WS ⊥ ST, then YT ⊥ TS
if WS || YT and WS ⊥ ST, then YT ⊥ TS
if WX || LM and YZ || LM and m∠WSX = 180 and m∠YTZ = 180 and m∠LRM = 180 and m∠STR = 180, then WX || YZ
if distance XY = distance ZW and XY || WZ, then XW || YZ
((m∠YWX) + (m∠WXY)) + (m∠XYW) = 180
if m∠ABC = m∠XYZ and m∠BCA = m∠YZX, then m∠CAB = m∠ZXY
if △XYZ is a right triangle, then (m∠YZX) + (m∠ZXY) = 90
if △XYZ is a right triangle and distance XY = distance YZ, then m∠YZX = 45
if △XYZ is an equilateral triangle, then m∠XYZ = 60
if △XYZ is an equilateral triangle, then m∠ZXY = 60
if quadrilateral WXYZ is convex, then (((m∠WXY) + (m∠XYZ)) + (m∠YZW)) + (m∠ZWX) = 360
if m∠XYZ = 180, then m∠WYZ = (m∠WXY) + (m∠YWX)
if m∠XYZ = 180, then (m∠WXY) + (m∠XWY) = m∠WYZ
if distance WY = distance XY and ray YP bisects ∠XYZ and m∠WYZ = 180, then WX || YP
if m∠BCA = m∠YZX and m∠CAB = m∠ZXY and distance AB = distance XY, then △CAB ≅ △ZXY
if m∠ABC = m∠XYZ and m∠CAB = m∠ZXY and distance AC = distance XZ, then △BAC ≅ △YXZ
if m∠PXY = 90 and m∠PZY = 90 and ray YP bisects ∠XYZ, then distance PX = distance PZ
if distance YX = distance YZ and XT ⊥ TZ and ZS ⊥ SX and m∠XSY = 180 and m∠ZTY = 180, then distance ZS = distance XT
if m∠YXZ = m∠XYZ, then distance ZX = distance ZY
if m∠ZXY = m∠ZYX, then distance ZX = distance ZY
if m∠ZXY = m∠ZYX, then distance XZ = distance YZ
if m∠ZYX = m∠YZX, then distance XY = distance XZ
if ∠ABC is a right angle and ∠XYZ is a right angle and distance AC = distance XZ and distance BC = distance YZ, then △ABC ≅ △XYZ
if PX ⊥ XY and PZ ⊥ ZY and distance XP = distance ZP, then m∠PYX = m∠PYZ
if ray AS bisects ∠CAB and ray BT bisects ∠ABC and m∠APS = 180 and m∠BPT = 180 and PZ ⊥ ZA and PX ⊥ XA and PX ⊥ XB and PY ⊥ YB and PZ ⊥ ZC and PY ⊥ YC, then ray CP bisects ∠BCA
if distance AB = x, then distance BA = x
if distance AB = distance CD, then distance AB = distance DC
if distance AB = distance CD, then distance BA = distance DC
if distance AB = distance CD, then distance DC = distance BA
if distance AB = distance CD, then distance DC = distance AB
if distance AB = distance CD, then distance CD = distance BA
if m∠ABC = m∠XYZ, then m∠ABC = m∠ZYX
if m∠ABC = m∠XYZ, then m∠CBA = m∠XYZ
if m∠ABC = m∠XYZ, then m∠CBA = m∠ZYX
if m∠ABC = m∠XYZ, then m∠ZYX = m∠CBA
if m∠ABC = m∠XYZ, then m∠XYZ = m∠CBA
if m∠ABC = m∠XYZ, then m∠ZYX = m∠ABC
if m∠ABC = x, then m∠CBA = x
if m∠ABC = 180, then distance AC = (distance AB) + (distance BC)
if m∠ABC = 180, then (distance AB) + (distance CB) = distance AC
if M is the midpoint of line AB, then (distance AM) + (distance MB) = distance AB
if m∠ABC = 180, then m∠BAX = m∠CAX
if m∠ABC = 180, then m∠CAX = m∠BAX
if m∠ABC = 180, then m∠ACX = m∠BCX
if m∠ABC = 180, then m∠XCB = m∠XCA
if m∠CBA = 180, then m∠XAC = m∠XAB
if m∠ABC = 180, then m∠XAC = m∠XAB
if m∠ABC = 180, then m∠XAB = m∠XAC
if m∠CBA = 180, then m∠XAC = m∠XAB
if m∠CBA = 180, then m∠XAB = m∠XAC
if m∠CBA = 180, then m∠ACX = m∠BCX
if m∠ABC = 180, then m∠XAB = m∠CAX
if m∠ADC = 180 and m∠BEC = 180, then m∠ACB = m∠DCE
if m∠ADC = 180 and m∠BEC = 180, then m∠BCA = m∠ECD
if ∠ABX and ∠XBC are supplementary, then m∠ABC = 180
if m∠ABC = 180, then (m∠ABX) + (m∠XBC) = 180
if m∠ABC = 180, then ∠ABX and ∠XBC are supplementary
if m∠DAB = m∠CBA and m∠AXB = 180, then m∠DAX = m∠CBX