2-5 Postulates Ms. Andrejko

Real World

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

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)

Postulates 2.1-2.7 Midpoint theorem

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.

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.

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.

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.

Examples In the figure, line m and 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

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