2.5 Postulates and Paragraph Proofs Objective: You will be able to identify and use basic postulates about points, lines, and planes and write paragraph proofs.

Postulate or Axiom Rules Describe fundamental relationships between geometric terms

Postulates 2.1 Through any two points, there is exactly one line. 2.2 Through any 3 points not on the same line, there is exactly one plane.

Postulates, Thms., Defintions 2.3 A line contains at least two points. 2.4 A plane contains at least three points not on the same line. 2.5 If two points lie in a plane, then the entire line containing those points lies in that plane. 2.6 If two lines intersect, then there intersection is exactly one point. 2.7 If two planes intersect, then their intersection is a line.

Thms and Def. NOT listed (write in notes) Definition of a midpoint: same as midpoint thm., used for anything with midpoints Definition of congruent segments: used to change congruency statements to equations Multiplication Property of Equality: When you multiply both sides of an EQUATION

If plane T contains contains point G, then plane T contains point G. Determine whether the following statement is always, sometimes, or never true. Use a postulate to explain. If plane T contains contains point G, then plane T contains point G. Answer: Always; Postulate 2.5 states that if two points lie in a plane, then the entire line containing those points lies in the plane. Example 5-2a

For , if X lies in plane Q and Y lies in plane R, then plane Q intersects plane R. Answer: Sometimes; planes Q and R can be parallel, and can intersect both planes. Example 5-2b

contains three noncollinear points. Answer: Never; noncollinear points do not lie on the same line by definition. Example 5-2c

a. Plane A and plane B intersect in one point. Determine whether each statement is always, sometimes, or never true. Explain. a. Plane A and plane B intersect in one point. b. Point N lies in plane X and point R lies in plane Z. You can draw only one line that contains both points N and R. Answer: Never; Postulate 2.7 states that if two planes intersect, then their intersection is a line. Answer: Always; Postulate 2.1 states that through any two points, there is exactly one line. Example 5-2d

c. Two planes will always intersect a line. Determine whether each statement is always, sometimes, or never true. Explain. c. Two planes will always intersect a line. Answer: Sometimes; Postulate 2.7 states that if the two planes intersect, then their intersection is a line. It does not say what to expect if the planes do not intersect. Example 5-2e