# Postulates & Theorems Study So Far

## 38 terms

### Postulate 1

Ruler Postulate: The points on a line can be paired in a one to one correspondence with the real numbers such that:

1. any two given points can have coordinates 0 and 1

2. the distance between two points is the absolute value of the difference of their coordinates

### Postulate 2:

If B is between A and C, then AB + BC = AC

### Postulate 3:

Protractor Postulate:

Given a point, X, on line PR, consider line XP and line XR, as well as all the other rays that can be drawn, with X as an endpoint, on one side of line PR. These rays can be paired with the real numbers from 0 to 180 such that:

1. line XP is paired with 0, and line XR is paired with 180
2. If line XA is paired with a number, c, and line XB is paired with a number, d, then m<AXB = |c-d|

### Postulate 4:

If point D is in the interior of <ABC, then m<ABD + m<DBC = m<ABC.

### Postulate 5

Through any two points there exists exactly one line.

### Postulate 6

Through any three noncollinear points there exists exactly one plane.

### Postulate 7

If two planes intersect, then their intersection is a line.

### Postulate 8

If two points lie in a plane, then the line containing the points lies in the plane.

### Postulate 9

A line contains at least 2 points. A plane contains at least 3 noncollinear points. Space contains at least 4 noncoplanar points.

### Postulate 10:

Parallel Postulate:
Through a point not on a line, there exists exactly one line through the point parallel to the line.

### Postulate 11:

Corresponding Angles Postulate:
If two parallel lines are cut by a transversal, then the corresponding angles formed are congruent.

### Postulate 12:

Converse of the Corresponding Angles Postulate:
If two lines are cute by a transversal and the corresponding angles formed are congruent, then the lines are parallel.

### Postulate 13:

Side-Side-Side (SSS) Triangle Congruence Postulate:
If three sides of one triangle are congruent to three sides of another triangle, then the triangles are congruent.

### Postulate 14:

The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs.

### Postulate 15:

Side-Angle-Side (SAS) Triangle Congruence Postulate:
If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.

### Postulate 16:

Angle-Side-Angle (ASA) Triangle Congruence Postulate:
If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the triangles are congruent.

### Postulate 17:

Parallel Lines Postulate:
If two lines are parallel, then they have the same slope. All vertical lines are parallel to each other.

### Postulate 18:

Perpendicular Lines Postulate:
If two non-vertical lines are perpendicular, then the product of their slopes is -1. Vertical and horizontal lines are perpendicular to each other.

### Postulate 19:

Area congruence Postulate:
If two polygons are congruent, then they have the same area.

### Postulate 20:

The area of a region is equal to the sum of the areas of its non-overlapping parts.

### Postulate 21:

Angle-Angle (AA) Triangle Similarity Postulate:
If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.

### Theorem 4-1

If two lines intersect, then they intersect at exactly one point.

### Theorem 4-2

If there is a line and a point not on the line, then exactly one plane contains them.

### Theorem 4-3

If two lines intersect, then there exists exactly one plane that contains them.

### Theorem 5-1

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

### Theorem 5-2

If two lines in a plane are perpendicular to the same line, then they are parallel to each other.

### Theorem 5-3

In a plane, if a line is perpendicular to one of two parallel lines, then it is perpendicular to the other one.

### Theorem 5-4

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

### Theorem 5-5

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

### Theorem 5-6:

Right Angles Theorem:
All right angles are congruent.

### Theorem 5-7:

Transitive Property of Parallel Lines:
If two lines are parallel to the same line, then they are parallel to each other.

### Theorem 6-1:

Congruent Complements Theorem:
If two angles are complementary to the same angle or to congruent angles, then tehy are congruent.

### Theorem 6-2:

Congruent Supplements Theorem:
If two angles are supplementary to the same angle or to congruent angles, then they are congruent.

### Theorem 6-3:

Linear Pair Theorem:
If two angles form a linear pair, then they are supplementary.

### Theorem 6-4:

Vertical Angles Theorem:
If two angles are vertical angles, then they are congruent.

### Theorem 6-5

If a point lies on the perpendicular bisector of a segment, then the point is equidistant from the endpoints of the segment.

### Theorem 6-6

If a point is equidistant from the endpoints of a segment, then the point lies on the perpendicular bisector of the segment.

### Theorem 6-7

If a point lies on the bisector of an angle, then the point is equidistant from the sides of the angle.