Postulates P.1
Postulate 1 - Ruler Postulate
1. The points on a line can be paired with the real numbers in such a way that any two points can have coordinates 0 and 1.
2. Once a coordinate system has been chosen in this way, the distance between any two points equals the absolute value of the difference of their coordinates.
Postulate 6
Through any two points there is exactly one line.
Theorem 1-1
If two lines intersect, then they intersect in exactly one point.
Theorem 2-3- Vertical angles theorem
Vertical 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.
Postulate 2- Segement addition postulate
If B is between A and C, the AB+BC = AC
Postulate 7
Through any three points there is at least one plane, and through any three noncollinear points there is exactly one plane.
Theorem 1-2
Through a line and a point not in the line there is exactly one plane.
Theorem 2-4
If two lines are perpendicular, then they form congruent adjacent angles.
Theorem 3-1
If two parallel planes are cut by a third plane, then the lines of intersection are parallel.
Postulate 3- Protractor Postulate
On line AB in a given plane, choose any point O between A and B. Consider OA and OB and all thee rays that can be drawn from O on one side of line AB. These rays can be paired with the real numbers from 0 to 180 in such a way that:
a. Ray OA is paired with 0, and ray OB with 180
b. If ray OP is paired with x, and ray OQ with y, then measure of angle POQ = |x-y|
Postulate 8
If two points are in a plane, then the line that contains the points is in that plane.
Theorem 1-3
If two lines intersect, then exactly one plane contains the lines.
Theorem 2-5
If two lines form congruent adjacent angles, then the lines are perpendicular.
Theorem 3-2
If two parallel lines are cut by a transversal, then alternate interior angles are congruent.
Postulate 4- Angle addition postulate
If point B lies in the interior of angle AOC, then measure of angle AOB + measure of angle BOC = measure of angle AOC.
Postulate 9
If two planes intersect, then their intersection is a line.
Theorem 2-1- Midpoint Theorem
If M is the midpoint of line segement AB, then AM = 1/2 AB and MB= 1/2 AB
Theorem 2-6
If the exterior sides of two adjacent acute angles are perpendicular, then the angles are complementary.
Theorem 3-3
If two parallel lines are cut by a transversal, then same-side interior angles are suplementary.
Postulate 5
A line contains at least two points: a plane contains at least three points not all in one line: space contains at least four points not all in one plane.
Postulate 10
If two parallel lines are cut by a transversal, then corresponding angles are congruent.
Theorem 2-2- Angle bisector theorem
If ray BX is the bisector of angle ABC, then measure of angle ABX = 1/2 measure of angle ABC and measure of angle XBC= 1/2 the measure of angle ABC.
Theorem 2-7
If two angles are supplements of congruent angles ( or of the same angle), then the two angles are congruent.
Theorem 3-4
If a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other one also.