Geometry Rules

Postulates and Theorems

Postulate 1:

Ruler Postulate: The points on a line can be paired with the real numbers in such a way that any two points cann have coordinates 0 and 1

Once a system has been chosen, the distance between any two points is equal to the absolute value of their difference

Postulate 2:

Segment Addition Postulate: If B is between A and C, then AB + BC = AC

Postulate 3:

Protractor Postulate: On in a given plane, choose any point O between A and B

Consider [OA> and [OB> and all rays that can be drawn from O on one side of

These rays can be paired with the real numbers from 0 to 180 in such a way that [OA> is paired with 0 and [OB> is paired with 180

Postulate 4:

Angle Addition Postulate: If point B lies in the interior of ∠AOC, then m∠AOB + m∠BOC = m∠AOC

If ∠AOC is a straight angle and B is any point not on , then m∠AOB + m∠BOC = 180

Postulate 5:

A line has at least two points

A plane has at least three points not on one line

A space has at least 4 points not all in one plane

Postulate 6:

Through any two points there is exactly one line

Postulate 7:

Through any three points there is at least one plane and through any three noncollinear points there is exactly one plane

Postulate 8:

If two points are in a plane, then the line that contains the points are in the plane

Postulate 9:

If two planes intersect, then their intersection is a line

Postulate 10:

If two parallel lines are cut by a transversal, then the corresponding angles are congruent

Theorem 1-1:

If two lines intersect, then it's in one point

Theorem 1-2:

Through a line and a point not in the line there's exactly one plane

Theorem 1-3:

If two lines intersect, then exactly one plane contains the lines

Theorem 2-1:

Midpoint Theorem: If M is the midpoint of [AB], then AM = 1/2AB and MB = 1/2AB

Theorem 2-2:

Angle Bisector Theorem: If [BX> is the bisector of ∠ABC, then m∠ABX = 1/2m∠ABC and m∠XBC = 1/2m∠ABC

Theorem 2-3:

Vertical Angles are congruent

Theorem 2-4:

If two lines are perpendicular, then they form congruent adjacent angles

Theorem 2-5:

If two lines form congruent adjacent angles, then the lines are perpendicular

Theorem 2-6:

If the exterior sides of two adjacent angles are perpendicular then the angles are complementary

Theorem 2-7:

If two angles are supplements of congruent angles (or of the same angle), then the two angles are congruent

Theorem 2-8:

If two angles are complements of congruent angles (or of the same angle), then the two angles are congruent

Theorem 3-1:

If two parallel planes are cut by a third plane, then the lines of intersection are parallel

Theorem 3-2:

If two parallel lines are cut by a transversal, then the alternate interior angles are congruent

Theorem 3-3:

If two parallel lines are cut by a transversal, then the same-side interior angles are supplementary

Theorem 3-4:

If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other one also