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Algebraic Properties

  • A postulate is a statement that is assumed true without proof.
  • A theorem is a true statement that can be proven.

postulate

 Let a , b and c be real numbers.

  • Reflexive Property - a=a (A quantity is congruent (equal) to itself.)
  • Symmetric Property - If a=b, then b=a
  • Transitive Property - If a=b and b=c , then a=c
  • Addition Postulate - If a=b , then a+c=b+c
  • Subtraction Postulate - If a=b , then ac=bc
  • Multiplication Postulate - If a=b , then ac=bc
  • Division Postulate - If a=b and c0 , then ac=bc
  • Substitution Postulate - If a=b , then a can be substituted for b in any expression.
  • Distributive Postulate - a(b+c)=ab+ac
  • A straight line contains at least two points.
  • If two lines intersect, the intersection is only one point.
  • If two planes intersect, the intersection is only one line.
  • A plane must contain at least three noncollinear points.

theorem

  • Congruence of Segments
    • Segment congruence is reflexive, symmetric, and transitive.
      • Reflexive - For any segment AB, ABAB is congruent to AB
      • Symmetric - If AB=CD , then CD=AB
      • Transitive - If AB=CD and CD=EF . then AB=EF
  • Congruent Angles
    • Angle congruence is reflexive, symmetric, and transitive.
      • Reflexive - For any A, A=A
      • Symmetric - If A=B , then B=A
      • Transitive - If A=B and B=C, then A=C
  • Right Angle Congruence
    • All right angles are congruent.
  • Congruent Supplements
    • If two angles are supplementary to the same angle, then they are congruent.
    • If two angles are supplementary to congruent angles, then they are congruent.
  • Congruent Complementary
    • If two angles are complementary to the same angle, then they are congruent.
    • If two angles are complementary to congruent angles, then they are congruent.
  • Vertical Angles Congruence
    • Vertical angles are always congruent.

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