Unit 3: Lines and Angles Definitions and Theorems

Axiom

A self-evident or accepted rule

Property

A characteristic or quality that is always true

The Reflexive Property

Any number is equal to itself

The Substitution Property

The value of the expression remains the same after the substitution or replacement of a variable or quantity in an equation or expression with another value

The Addition Property

If you add the same number to both sides in an equation, the equality remains true

The Subtraction Property

If you subtract the same number to both sides in an equation, the equality remains true

The Multiplication Property

If you multiply the same number to both sides in an equation, the equality remains true.

The Division Property

If you divide the same number to both sides in an equation the equality remains true

The Triangle Angle Sum Theorem

The sum of the angles of a triangle is 180º

Corollary

A theorem that can be easily proved as a consequence of a postulate or another theorem.

Postulate 1

Two points determine a line

Postulate 2

Three noncollinear points determine a plane

Coordinate

The location of a point

Postulate 3 The Ruler Postulate

The points on a line can be numbered so that positive number differences measure distances.

Postulate 4 The Protractor Postulate

In a plane, any two opposite rays can be paired with real numbers 0-180 so that positive number differences measure angles

Theorem 1 The Betweenness of Points Theorem

If a point B is between two other points A and C on a line, then the distance from A to B added to the distance from B to C is equal to the total distance from A to C

Theorem 2 The Betweenness of Rays Theorem

When considering rays originating from a common point, if one ray is "between" two others, the sum of the smaller angles equals the larger angle

Midpoint

The "middle point" of a line segment that divides the line segment into two equal segments.

Corollary to the Ruler Postulate

A line segment has exactly one midpoint

Corollary to the Protractor Postulate

An angle has exactly one ray that bisects it

Complementary Angles

Two angles whose sum is 90º

Supplementary Angles

Two angles whose sum is 180º

Theorem 3

Complements of the same angle are equal

Theorem 4

Supplements of the same angle are equal

Opposite Rays

Two rays that point in opposite directions

Linear Pair

A pair of adjacent, supplementary angles

Vertical Angle

A pair of angles that share the same vertex and are on opposite sides of two intersecting straight lines

Theorem 5

The angles in a linear pair are supplementary

Theorem 6

Vertical angles are equal

Theorem 7

Perpendicular lines form right angles

Theorem 8

If the angles in a linear pair are equal, then their sides are perpendicular