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Algebraic Properties

on . Posted in Algebra

  • 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 \(\;a - c = b - c\; \)
  • Multiplication Postulate - If \(\;a = b\; \) , then \(\;ac = bc\; \)
  • Division Postulate - If \(\;a = b\; \) and \(\;c \ne 0\; \) , then \(\; \frac {a}{c} = \frac {b}{c}\; \)
  • Substitution Postulate - If \(\;a = b\; \) , then \(\;a\; \) can be substituted for \(\;b\; \) in any expression.
  • Distributive Postulate - \(\;a \left (b + c \right ) = ab + ac\; \)
  • 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 \(\;AB\; \), \(\;AB\; \)AB 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 \(\; \angle A\; \), \(\; \angle A\; = \angle A \)
      • Symmetric - If \(\; \angle A = \angle B \; \) , then \(\; \angle B = \angle A \; \)
      • Transitive - If \(\; \angle A = \angle B \; \) and \(\; \angle B = \angle C \; \), then \(\; \angle A = \angle C \; \)
  • 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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