2-5 Postulates

1. 2-5 Postulates Ms. Andrejko

2. Real World

3. Vocabulary • Postulate/Axiom- is a statement that is accepted as true without proof • Proof- a logical argument in which each statement that you make is supported by a postulate or axiom • Theorem- a statement that has been proven that can be used to reason • Deductive Argument- forming a logical chain of statements linking the given to what you are trying to prove

4. Steps to a proof • 1. List the given information and if possible, draw a diagram • 2. State the theorem or conjecture to be proven. • 3. Create a deductive argument • 4. Justify each statement with a reason (definition, algebraic properties, postulates, theorems) • 5. State what you have proven (conclusion)

5. Postulates • 2.1-2.7 • Midpoint theorem

6. Examples • Explain how the figure illustrates that each statement is true. Then state the postulate that can be used to show each statement is true. • The planes J and K intersect at line m. • The lines l and m intersect at point Q. Postulate: If 2 planes intersect, then their intersection is a line. Postulate: If 2 lines intersect, then their intersection is exactly one point.

7. Practice • Explain how the figure illustrates that each statement is true. Then state the postulate that can be used to show each statement is true. • Line p lies in plane N. • Planes O and M intersect in line r. Postulate: If 2 points lie in a plane, then the entire line containing those points lies in that plane. Postulate: If 2 planes intersect, then their intersection is a line.

8. Examples • Determine whether each statement is always, sometimes, or never true. Explain your reasoning. • The intersection of two planes contains at least two points. • If three planes have a point in common, then they have a whole line in common. ALWAYS. The intersection of 2 planes is a line, and we must have at least 2 points in order to create a line. SOMETIMES. 3 planes can intersect at the same line which contains the same point, but they don’t have to.

9. Practice • Determine whether each statement is always, sometimes, or never true. Explain your reasoning. • Three collinear points determine a plane • Two points A and B determine a line • A plane contains at least three lines NEVER. Postulate tells us that we must have 3 noncollinear points ALWAYS. You can always create a line through any 2 points. SOMETIMES. A plane may contain 3 lines, but it doesn’t have to contain any lines in order to be a plane.

10. Examples • In the figure, line mand lie in plane A. State the postulate that can be used to show that each statement is true. • Points L, and T and line m lie in the same plane. • Line m and intersect at T. 2.5: If 2 points lie in a plane, then the entire line containing those points lies in that plane 2.6: If 2 lines intersect, then their intersection is exactly one point

11. Practice • In the figure, and are in plane J and pt. H lies on State the postulate that can be used to show each statement is true. • G and H are collinear. • Points D, H, and P are coplanar. 2.3: A line contains at least 2 points. 2.2: Through any 3 noncollinear points, there is exactly one plane