Theorems and Postulates

Postulates

1. Any segment or angle is congruent to itself. (Reflexive)

2. A line segment is the shortest path between two points.

3. Two points determine a line (or ray or segment)

4. Through a point not on a line there is exactly one parallel to the given line. (Parallel Postulate)

5. Three noncollinear points determine a plane

6. If  a line intersects a plane not containing it, then the intersection is exactly one point.

7. If two planes intersect, their intersection is exactly one line.

Theorems

1. If a segment is added to two congruent segments, the sums are congruent. (Addition Property)

2. If congruent segments are added to congruent segments, the sums are congruent.

3. If a segment is subtracted from congruent segments, the differences are congruent. (Subtraction Property)

4. If segments are congruent, their like divisions are congruent. (Division Property)

5. If segments are congruent, their like multiples are congruent. (Multiplication Property)

6. If two points are each equidistant from the endpoints of a segment, then the two points determine the perpendicular bisector of that segment.

7. If two nonvertical lines are parallel, then their slopes are equal.

8. If the slopes of two nonvertical lines are equal, then the lines are parallel.

9. If two lines are perpendicular and neither is vertical, each line's slope is the opposite reciprocal of the other's.

10. If a line's slope is the opposite reciprocal of another line's slope, the two lines are perpendiculer.

11. If two parallel lines are cut by a transversal, each pair of interior angles on the same side of the transversal are supplementary.

12. If two parallel lines are cut by a transversal, each pair of exterior angles on the same side of the transversal are supplementary.

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

14. If two lines are parallel to a third line, they are parallel to each other, (Transitive Property of Parallel Lines)

15. A line and a point not on the line determine a plane.

16. Two intersecting lines determine a plane.

17. Two parallel lines determine a plane.

18. If a line is perpendicular to two distinct lines that lie in a plane and that pass through its foot, then it is perpendicular to the plane,

19. If a plane intersects two parallel planes, the lines of intersection are parallel.

20. If three or more parallel lines are intersected by two transversals, the parallel lines divide the transversals proportionally.

21.