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
(a + b) + c = (b + a) + c
(a + b) + c = (c + a) + b
(c + a) + b = (a + b) + c
(a + b) - c = (a - c) + b
((a + b) + c) + ((d + e) + f) = ((((a + b) + c) + d) + e) + f
((((a + b) + c) + d) + e) + f = ((((a + e) + b) + d) + f) + c
((((a + b) + c) + d) + e) + f = ((a + b) + (c + d)) + (e + f)
(0 + a) + b = a + b
((a + b) + c) + d = ((a + c) + b) + d
(a ⋅ (b ⋅ (-1))) ⋅ 2 = (a ⋅ b) ⋅ (-2)
(a ⋅ (-1)) ⋅ (a ⋅ (-1)) = a ⋅ a
(a ⋅ (1 / 2)) + (a ⋅ (1 / 2)) = a
a + a = a ⋅ 2
(a + a) + a = a ⋅ 3
((a + a) + a) + a = a ⋅ 4
((a + b) + a) + b = (a + b) ⋅ 2
(a + b) ⋅ (a + b) = ((a ⋅ a) + ((a ⋅ b) ⋅ 2)) + (b ⋅ b)
(b ⋅ d) ⋅ (a / b) = d ⋅ a
(b ⋅ d) ⋅ (c / d) = b ⋅ c
(0 + (a ⋅ 2)) / 2 = a
(a ⋅ 2) ⋅ (1 / 2) = a
((b ⋅ 2) + (a ⋅ 2)) / 2 = b + a
((a ⋅ b) ⋅ 2) + (((a ⋅ a) + ((a ⋅ b) ⋅ (-2))) + (b ⋅ b)) = (a ⋅ a) + (b ⋅ b)
(a + (b ⋅ (-1))) ⋅ (a + (b ⋅ (-1))) = ((a ⋅ a) + ((a ⋅ b) ⋅ (-2))) + (b ⋅ b)
if M is the midpoint of line XY and ∠ZMY is a right angle, then △XMZ ≅ △YMZ
if M is the midpoint of line AB and m∠XMY = 180 and ∠MAY is a right angle and ∠MBX is a right angle, then △YMA ≅ △XMB
if distance AX = distance XB and M is the midpoint of line AB, then △AXM ≅ △BXM
if ray BX bisects ∠ABC and distance AB = distance CB, then △ABX ≅ △CBX
if m∠ABC = 180 and m∠DCB = 180 and ∠XAC is a right angle and ∠YDB is a right angle and distance XA = distance YD and distance AB = distance DC, then distance XC = distance YB
if distance AX = distance BX, then m∠BAX = m∠ABX
if distance XZ = distance YZ, then m∠ZXY = m∠ZYX
if distance XZ = distance YZ, then m∠ZXY = m∠XYZ
if distance ZX = distance ZY, then m∠ZXY = m∠ZYX
if distance ZY = distance ZX, then m∠ZXY = m∠XYZ
if distance YZ = distance YE, then m∠ZEY = m∠EZY
if distance XY = distance YZ, then m∠YZX = m∠ZXY
if distance XY = distance YZ, then m∠ZXY = m∠XZY
if distance XZ = distance YZ, then m∠XYZ = m∠YXZ
if m∠WXY = 180 and m∠XYZ = 180 and distance WX = distance ZY and distance XM = distance YM, then distance WM = distance ZM
if S is the midpoint of line XY and T is the midpoint of line XZ and distance XY = distance XZ, then distance YT = distance ZS
if distance XS = distance YS and M is the midpoint of line XY, then m∠XMS = 90
if SM ⊥ MY and M is the midpoint of line XY, then distance SX = distance SY
if A is the midpoint of line XY and B is the midpoint of line ZX and PB ⊥ BX and PA ⊥ AY, then distance PZ = distance PY
if △XYZ is an equilateral triangle and M is the midpoint of line XY, then ∠XMZ is a right angle
if △XYZ is an equilateral triangle, then m∠XYZ = m∠YXZ
if △XYZ is an equilateral triangle, then m∠ZXY = m∠XZY
if △XYZ is an equilateral triangle, then m∠YZX = m∠ZXY
if △XYZ is an equilateral triangle, then m∠XYZ = m∠YZX
if △XYZ is an equilateral triangle, then m∠ZXY = m∠XYZ
if △XYZ is an equilateral triangle, then m∠XYZ = m∠ZXY
if WXYZ is a parallelogram, then △ZWY ≅ △XYW
if WXYZ is a parallelogram, then distance WZ = distance XY
if WXYZ is a parallelogram, then distance ZW = distance YX
if WXYZ is a parallelogram, then distance WX = distance YZ
if WXYZ is a parallelogram, then distance WX = distance ZY
if WXYZ is a parallelogram and m∠WPY = 180 and m∠XPZ = 180, then △PYZ ≅ △PWX
if WXYZ is a parallelogram and m∠WPY = 180 and m∠XPZ = 180, then distance ZP = distance XP
if distance WX = distance YZ and distance WZ = distance XY, then XWZY is a parallelogram
if distance WX = distance ZY and distance WZ = distance XY, then WXYZ is a parallelogram
if distance WX = distance YZ and distance XY = distance ZW, then WXYZ is a parallelogram
if WX || ZY and distance WX = distance YZ, then WXYZ is a parallelogram
if m∠XYZ = m∠ZWX and m∠WXY = m∠YZW, then WXYZ is a parallelogram
if m∠YZW = m∠WXY and m∠ZWX = m∠XYZ, then WXYZ is a parallelogram
if distance WP = distance PY and distance XP = distance PZ and m∠WPY = 180 and m∠XPZ = 180, then WXYZ is a parallelogram
if WXYZ is a parallelogram and m∠WSX = 180 and m∠ZTY = 180 and distance WS = distance TY, then ZS || TX
if WXYZ is a rectangle, then ∠ZWX is a right angle
if WXYZ is a rectangle, then ∠XYZ is a right angle
if WXYZ is a rectangle, then ∠YZW is a right angle
if WXYZ is a rectangle, then distance WY = distance XZ
if m∠WXY = m∠XYZ and m∠XYZ = m∠YZW and m∠YZW = m∠ZWX, then WXYZ is a rectangle
if WXYZ is a parallelogram and distance WY = distance XZ, then WXYZ is a rectangle
if WXYZ is a parallelogram and m∠YZW = m∠ZWX, then WXYZ is a rectangle
if WXYZ is a rhombus, then distance WZ = distance ZY
if WXYZ is a rhombus, then distance YZ = distance ZW
if WXYZ is a rhombus, then distance ZW = distance WX
if WXYZ is a rhombus, then distance XY = distance YZ
if WXYZ is a rhombus and m∠WPY = 180 and m∠XPZ = 180, then △PWX ≅ △PYZ
if WXYZ is a rhombus, then m∠WZX = m∠YZX
if WXYZ is a rhombus, then m∠XYW = m∠ZYW
if distance WX = distance XY and distance XY = distance YZ and distance YZ = distance ZW, then WXYZ is a rhombus
if WXYZ is a parallelogram and ∠YPZ is a right angle and m∠WPY = 180 and m∠XPZ = 180, then WXYZ is a rhombus
if WXYZ is a rhombus and m∠WXY = 60, then △YZW is an equilateral triangle
if WXYZ is a square, then distance XY = distance YZ
if WXYZ is a square, then distance ZW = distance WX
if WXYZ is a square, then distance YZ = distance ZW
if WXYZ is a square, then WXYZ is a rhombus
if WXYZ is a rhombus and ∠WXY is a right angle, then WXYZ is a square
if distance WX = distance XY and distance XY = distance YZ and distance YZ = distance ZW and ∠WXY is a right angle, then WXYZ is a square
if ABCD is a square, then m∠BCA = 45
if quadrilateral WXYZ is an isosceles trapezoid, then m∠ZWX = m∠WXY
if quadrilateral WXYZ is an isosceles trapezoid, then m∠ZWX = m∠YXW
if m∠ZWX = m∠YXW and WX || ZY, then distance WZ = distance XY
if quadrilateral WXYZ is an isosceles trapezoid, then distance WY = distance XZ
if quadrilateral WXYZ is a trapezoid and distance WY = distance XZ, then distance ZW = distance YX
if the y coordinate of point Z = b and the y coordinate of point Y = b and the y coordinate of point W = 0 and the y coordinate of point X = 0 and S is the midpoint of line WZ and T is the midpoint of line XY and not((the x coordinate of point T) - (the x coordinate of point S) = 0) and not((the x coordinate of point X) - (the x coordinate of point W) = 0), then ST || WX
if b > 0, then (0 - 0) / ((b ⋅ 2) - 0) = 0
if X is the midpoint of line RT and Y is the midpoint of line ST, then RS || XY
if X is the midpoint of line RT and Y is the midpoint of line ST, then (distance XY) ⋅ 2 = distance RS
if m∠CAB = m∠ZXY and m∠ABC = m∠XYZ, then △ABC ∼ △XYZ
if m∠XYZ = m∠XPY and m∠YXZ = m∠PXY, then △XYZ ∼ △XPY