1. Postulates and Paragraph Proofs
Postulate (Axiom) – A statement that describes a fundamental relationship between the basic terms of geometry. The basic ideas about points, lines, and planes can be stated as postulates.
Postulate 2.1
Through any two points, there is exactly one line.

2. Postulates and Paragraph Proofs
Through any three points not on the same line, there is exactly one plane.
Postulate 2.3
A line contains at least two points.
Postulate 2.4
A plane contains at least three points not on the same line.

3. Postulates and Paragraph Proofs
If two points lie in a plane, then the entire line containing those points lies in that plane.
Postulate 2.6
If two lines intersect, then their intersection is exactly one point.
Postulate 2.7
If two planes intersect, then their intersection is a line.

4. 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

5. 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

6. Postulates and Paragraph Proofs
Theorem – A statement or conjecture that can be proven to be true.
Proof – A logical argument in which each statement you make is supported by a statement that is accepted as true.
Theorem 2.8 – Midpoint Theorem
If M is the midpoint of segment AB, then segment AM is congruent to segment MB.

7. Postulates and Paragraph Proofs
Five essential parts of a proof:
State the theorem or conjecture to be proven.
List the given information.
If possible, draw a diagram to illustrate the given information.
State what is to be proved.
Develop a system of deductive reasoning.

8. Given is the midpoint of and X is the midpoint of write a paragraph proof to show that
Example 5-3b

9. Proof: We are given that S is the midpoint of and X is the midpoint of By the definition of midpoint, Using the definition of congruent segments, Also using the given statement and the definition of congruent segments, If then Since S and X are midpoints, By substitution, and by definition of congruence,
Example 5-3c