# 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.