# 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 \(\;AB = EF\; \)

- Segment congruence is reflexive, symmetric, and transitive.
- 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 \(\; \angle A = \angle C \; \)

- Angle congruence is reflexive, symmetric, and transitive.
- 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.