1 2-5 Postulates andParagraph Proofs

2 What is a Postulate? A Postulate or axiom is a statement that is accepted as fact.

3 Postulates: Points, Lines, and Planes

4 Intersections of Lines and Planes Postulate 2.6 lines If two lines intersect, then they intersect point in exactly one point. Postulate 2.7 planes If two planes intersect, then they intersect line in exactly one line.

5 Example 1 Identify the postulate that belongs to each statement. Line m contains points F and G. Point E can also be on line m. Lines s and t intersect at point D. Points A, B, and C are noncollinear. Points A, B, and C determine a plane. Planes P and Q intersect in line m A line contains at least two points If two lines intersect, then their intersection is a point. Three noncollinear points determine a plane The intersection of two planes is a line.

6 Analyze Statements Using Postulates Determine whether each statement is always, sometimes, or never true. Points A and B are collinear. Line r contains only point P. Four points are noncollinear If two coplanar lines intersect, then the point of intersection lies in the same plane as the two lines. Always Never Sometimes Always

7 A postulate is a statement of “fact” A Theorem is a statement that has been “proven” true Proof: a logical argument in which each step towards the conclusion is supported by a definition, postulate, or theorem i.e. supporting statements. Paragraph Proof: a type of proof that gives the steps and supporting statements in paragraph form Informal Proof: paragraph proofs are a type of informal proof

8 Theorem 2-1 Midpoint Theorem If M is the midpoint of AB, then AM = MB ABM

9 Proofs We will take a closer look at proofs in the next section.

10 Have a great day!!