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 $\;a - c = b - c\;$
• 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.