Section 2.4 In Geometry, rules that are accepted without proof are called postulates or axioms. Rules that are proved are called theorems. Postulates and theorems are often written as conditional statements. Unlike the converse of a definition, the converse of a postulate or theorem cannot be assumed to be true.

Postulates

Points Postulate 1. A line contains at least two points. 2. A plane contains at least 3 noncollinear points. M t

Line Postulate Through any two points there exists exactly one line.

Intersection of Lines Postulate If two lines intersect, then their intersection is a point. a b

Plane Postulate Through any three noncollinear points there is exactly one plane.

Flat Plane Postulate If two points lie in a plane, then the line containing them lies in the plane. S

Intersection of Planes Postulate If two planes intersection then their intersection is a line.

Algebraic Connection 1. One way to graph a linear equation is to plot two points whose coordinates satisfy the equation and then connect them with a line. (Line Postulate) 2. One way to find a common solution of two linear equations is to graph the lines and find the coordinates of their intersection. (Line Intersection Postulate)

Example 1 State the postulate illustrated by the diagram. Plane Postulate

Example 2 Use the diagram to write examples of Part 2 of the Points Postulate and the Plane Intersection Postulate. Part 2 of the Points Postulate: Plane Q contains points V, Y, and W. Plane Intersection Postulate: The intersection of plane Q and plane P is line b.

Interpreting a Diagram When you interpret a diagram, you can assume information about size or measure only if it is marked.

1. Can assume all points are coplanar 1. Can assume all points are coplanar. Cannot assume points E, F, and G are collinear. 2. Can assume CJD and DJE are a linear pair. Cannot assume lines CE and BF intersect. 3. Can assume CJD and HJE are vertical angles. Cannot assume lines CE and BF do not intersect.

4. Can assume points A, H, J, and D are collinear 4. Can assume points A, H, J, and D are collinear. Cannot assume CJD  BHJ. 5. Can assume lines CE and AD intersect at point J. Cannot assume lines CE and AD are perpendicular or mCJD = 90°.

Example 3

A line is a line perpendicular to a plane if and only if the line intersects the plane in a point and is perpendicular to every line in the plane that intersects it at that point. K

Example 4 Which of the following statements cannot be assumed from the diagram? 1. E, D, and C are collinear. 2. The intersection of line BD and line EC is D. #3 and #4