Function Angle

The measurement of angle ABC (m∠ABC) is the magnitude of the smallest rotation that maps line BA to line BC

Here is an example:

X Z Y

The measurement of angle XYZ is 90°. Some definitions also allow this angle to be 270° or -270°, but we only allow an angle to have a single measurement with the smallest magnitude.

Format:

m∠ABC

Input:

point A -
point B -
point C -

Output:

number - None

Details:

A B C

Properties that reference this function:

m∠ABC = m∠CBA (Angle Symmetry Property)

Conditional properties that reference this function:

  • if m∠ABC = 180, then point B is in segment AC (link)
  • if m∠ABC = 180, then point X lies in interior of ∠ABC (link)
  • if (point M is in segment BC) and (m∠AME = 180) and (m∠ACD = 180), then point E lies in interior of ∠BCD (link)
  • if ((m∠ABX) + (m∠XBC) = 180) or ((m∠ABX) + (m∠XBC) < 180), then point X lies in interior of ∠ABC (link)
  • if point X lies in interior of ∠ABC, then ((m∠ABX) + (m∠XBC) < 180) or ((m∠ABX) + (m∠XBC) = 180) (link)
  • if ((m∠ABX) + (m∠XBC) < 180) or ((m∠ABX) + (m∠XBC) = 180), then m∠ABC = (m∠ABX) + (m∠XBC) (link)
  • if ∠ABC is a right angle, then m∠ABC = 90 (link)
  • if m∠ABC = 90, then ∠ABC is a right angle (link)
  • if ray BD bisects ∠ABC, then m∠ABD = m∠DBC (link)
  • if m∠ABD = m∠DBC, then ray BD bisects ∠ABC (link)
  • if ray BD bisects ∠ABC, then m∠DBC = (m∠ABC) / 2 (link)
  • if (ray BD bisects ∠ABC) and (m∠BPD = 180) and (PM ⊥ MB) and (PN ⊥ NB), then distance PM = distance PN (link)
  • if ∠ABC is an acute angle, then m∠ABC < 90 (link)
  • if m∠ABC = 180, then (distance AB) + (distance BC) = distance AC (link)
  • if m∠ABC = 180, then m∠BCX = m∠ACX (link)
  • if m∠ABC = 180, then m∠ABC = (m∠ABX) + (m∠XBC) (link)
  • if (distance AM = distance MB) and (m∠AMB = 180), then M is the midpoint of line AB (link)
  • if M is the midpoint of line AB, then m∠AMB = 180 (link)
  • if ∠ABC and ∠DEF are complementary, then (m∠ABC) + (m∠DEF) = 90 (link)
  • if ∠ABC and ∠DEF are supplementary, then (m∠ABC) + (m∠DEF) = 180 (link)
  • if (m∠ABC) + (m∠DEF) = 180, then ∠ABC and ∠DEF are supplementary (link)
  • if m∠CDE = m∠CAB, then DE || AB (link)
  • if (AB || YZ) and (m∠ABC = 180), then BC || YZ (link)
  • if (AB || YZ) and (m∠AXB = 180), then AX || YZ (link)
  • if (AB || YZ) and (m∠ABC = 180) and (m∠XYZ = 180), then AC || XZ (link)
  • if (AB || XY) and (m∠ABC = 180) and (m∠XYZ = 180), then AC || XZ (link)
  • if (AC || XZ) and (m∠ABC = 180) and (m∠XYZ = 180), then AB || YZ (link)
  • if (AC || XZ) and (m∠ABC = 180) and (m∠XYZ = 180), then AB || XY (link)
  • if (AB || CD) and (m∠AXB = 180) and (m∠CYD = 180), then XB || CY (link)
  • if (AB || CD) and (m∠AXY = 180) and (m∠XYB = 180), then XY || CD (link)
  • if m∠abc = 90, then area of △ABC = (((distance AB) ⋅ (distance BC)) ⋅ 1) / 2 (link)
  • if AB || CD, then (m∠BAC) + (m∠ACD) = 180 (link)
  • if (m∠BAC) + (m∠ACD) = 180, then AB || CD (link)
  • if (m∠DBA) + (m∠CDB) = 180, then AB || CD (link)
  • if (distance AB = distance DE) and (m∠ABC = m∠DEF) and (distance BC = distance EF), then area of △ABC = area of △DEF (link)
  • if (distance AB = distance DE) and (m∠ABC = m∠DEF) and (distance BC = distance EF), then △ABC ≅ △DEF (link)
  • if (m∠ABC = m∠DEF) and (distance BC = distance EF) and (m∠BCA = m∠EFD), then △ABC ≅ △DEF (link)
  • if (m∠ABC = m∠DEF) and (distance BC = distance EF) and (m∠BCA = m∠EFD), then area of △ABC = area of △DEF (link)
  • if △ABC ≅ △DEF, then m∠ABC = m∠DEF (link)
  • if △ABC ≅ △DEF, then m∠CAB = m∠FDE (link)
  • if △ABC ≅ △DEF, then m∠BCA = m∠EFD (link)
  • if (distance AB = distance BC) and (distance BC = distance CD) and (distance CD = distance DA) and (m∠ABC = 90), then ABCD is a square (link)
  • if m∠ABC = 180, then area of △ACD = area of quadrilateral ABCD (link)
  • if m∠ABC = 180, then area of quadrilateral XABC = area of △XAC (link)
  • if m∠ABC = 180, then area of pentagon ACDEF = area of hexagon ABCDEF (link)
  • if m∠DEF = 180, then area of hexagon ABCDEF = area of pentagon ABCDF (link)
  • if (m∠ABC = 90) and (m∠DEF = 180), then m∠ABC < m∠DEF (link)
  • if line AB intersects line CD at point X, then m∠AXB = 180 (link)
  • if line AB intersects line CD at point X, then m∠CXD = 180 (link)
  • if (m∠ABD = 180) and (m∠ACD = 180), then m∠ABC = 180 (link)
  • if (m∠ABD = 180) and (m∠ACD = 180), then m∠BCD = 180 (link)
  • if (m∠ABC = 180) and (m∠BCD = 180), then m∠ABD = 180 (link)
  • if (m∠ABC = 180) and (m∠BCD = 180), then m∠ACD = 180 (link)
  • if (m∠ABC = m∠BCA) and (m∠BCA = m∠CAB), then △ABC is an equilateral triangle (link)
  • if (m∠ABC = m∠DEF) and (m∠BCA = m∠EFD) and (m∠CAB = m∠FDE) and ((distance AB) / (distance DE) = (distance BC) / (distance EF)) and ((distance BC) / (distance EF) = (distance CA) / (distance FD)), then △ABC ∼ △DEF (link)
  • if △ABC ∼ △DEF, then m∠ABC = m∠DEF (link)
  • if △ABC ∼ △DEF, then m∠BCA = m∠EFD (link)
  • if △ABC ∼ △DEF, then m∠CAB = m∠FDE (link)
  • if (points A B and C are collinear) and (distance AC > distance BC), then m∠ABC = 180 (link)
  • if (m∠ABC = m∠CDA) and (m∠BCD = m∠DAB), then quadrilateral ABCD is convex (link)
  • if ∠ABC is a right angle, then tangent of (m∠BCA) = (distance AB) / (distance BC) (link)
  • if ∠ABC is a right angle, then sine of (m∠BCA) = (distance AB) / (distance AC) (link)
  • if ∠ABC is a right angle, then cosine of (m∠BCA) = (distance BC) / (distance AC) (link)


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