Math

• A mathematical system is a set with one or more binary operations defined on it
• A binary operation is a rule that assigns to 2 elements of a set a unique third element
• Properties of Mathematical Systems


  • Closure
  • Identity Element
  • Inverse
  • Commutative
  • Associative
  • Distributive
• Postulates of Equality


  • Reflexive Property
  • Symmetric Property
  • Transitive Property
  • Substitution Property
• Properties of Congruence


  • Reflexive Property
  • Symmetric Property
  • Transitive Property
• Undefined terms/primitive terms


  • Point
  • Line
  • Plane
• Defined terms


  • Line segment
  • Ray
  • Angle
• Euclid's Postulates


  • Two points determine a line segment
  • A line segment can be extended indefinitely along a line
  • All right angles are congruent
• Point-Line-Plane Postulates


  • Unique Line Assumption
  • Number Line Assumption
  • Distance Assumption
  • If two points lie on a plane, the line containing them also lies on the plane
  • Through three non-collinear points, there is exactly one plane
• Theorems


  • Line Intersection Theorem
  • Betweenness Theorem
• Theorems are statements that can be deduced and proved from definitions, postulates, and previously proved theorems
• Plane


  A flat surface that extends indefinitely in all directions
• Line


  A straight path that extends indefinitely in both directions
• Point


  A specific location on a plane or line
• If two points lie on a plane, the line containing them also lies on the plane
• Through three non-collinear points, there is exactly one plane
• If TX = BK, then BK = TX
• 8(m + n) = 8m + 8n
• If CT = 12 and PR + CT = 20, then PR + 12 = 20
• m∠HIT ≅ m∠HIT
• If ∠S ≅ ∠P, ∠B ≅ ∠S, then ∠P ≅ ∠B
• Reflexive Property of Equality
• Symmetric Property of Equality
• Transitive Property of Equality
• Reflexive Property of Congruence
• Symmetric Property of Congruence
• ∠ABC ≅ ∠ABC
• If m∠B = m∠D and m∠D = m∠F, then m∠B = m∠F
• If GH = JK, then JK = GH
• If AB = CD,AB=9 then 9=CD
• If m∠C = 90, then 2(m∠C) + 15 = 2(90°) + 15°
• Plane P or Plane ABC


  Plane containing non-collinear points A, B and C
• In Geometry, a line is straight. It is impossible to have two points as point of intersections
• Three rays on line BC
• Congruent Triangles
• Definition of Congruent Triangles


  Two triangles are congruent if their corresponding parts are congruent
• ∆ABC ≅ ∆DEF
• ∠A ≅ ∠D, ∠B ≅ ∠E, ∠C ≅ ∠F, AB ≅ DE, BC ≅ EF, AC ≅ DF
• ∆ABE ≅ ∆DCE
• ∠A ≅ ∠D, ∠B ≅ ∠C, ∠E ≅ ∠E, AB ≅ DC, AE ≅ DE, BE ≅ CE