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Introduction to basic postulates and theorems of points, lines, and planes.

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- 1. 09/27/07 Warm-up <ul><li>Which of the following four statements can’t you conclude from the diagram? </li></ul><ul><li>A, B, and C are collinear. </li></ul><ul><li>B is the midpoint of AC. </li></ul><ul><li> DBC is a right angle. </li></ul><ul><li>E is in the interior of DBA </li></ul>D C B A E
- 2. Postulate vs. Theorem <ul><li>Postulate (axiom): </li></ul><ul><ul><li>A statement that is accepted as true without proof. </li></ul></ul><ul><li>Theorem: </li></ul><ul><ul><li>An important statement that must be proved before it can be accepted. </li></ul></ul>
- 3. Reading postulates and theorems <ul><li>Read carefully. </li></ul><ul><li>Reread each phrase, one at a time. </li></ul><ul><li>Look up any words you do not understand. </li></ul><ul><li>Try to identify conditions. </li></ul><ul><li>Look for key words, if , if and only if , exactly one , exists , unique , etc . </li></ul><ul><li>Visualize, draw diagrams, or model the situation </li></ul><ul><li>Imagine if it wasn’t true. What would that look like? </li></ul>
- 4. Postulates <ul><li>A line contains at least two points; </li></ul><ul><li>space contains at least four points not all in one plane. </li></ul><ul><li>A plane contains at least three points not all in one line; </li></ul>
- 5. More Postulates <ul><li>Through any two points there is exactly one line. </li></ul>
- 6. More Postulates <ul><li>and through any three noncollinear points there is exactly one plane. </li></ul><ul><li>Through any three points there is at least one plane, </li></ul>
- 7. More Postulates <ul><li>If two points are in a plane, then the line that contains the points is in that plane. </li></ul>
- 8. More Postulates <ul><li>If two planes intersect, then their intersection is a line. </li></ul>
- 9. p. 24: State a postulate, or part of a postulate, that justifies your answer to each exercise. <ul><li>Name two points that determine line l. </li></ul><ul><li>Name three points that determine plane M. </li></ul><ul><li>Name the intersection of planes M and N. </li></ul><ul><li>Does AD lie in plane M? </li></ul><ul><li>Does plane N contain any points not on AB? </li></ul>M N B A l C
- 10. Assignment A14 p. 25 #5-11
- 11. Theorems <ul><li>If two lines intersect, then they intersect in exactly one point. </li></ul>
- 12. Theorems <ul><li>Through a line and a point not in the line there is exactly one plane. </li></ul>
- 13. Theorems <ul><li>If two lines intersect, then exactly one plane contains the lines. </li></ul>

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The theorem you state refers to distance between the two points, or length of the segment having the two points as endpoints. All distance will be positive and real (not just positive integers or whole numbers, they can be rational or irrational). You can explain using a number line. You can have students demonstrate by uisng strips of paper to represent the distance between two points. Have students measure, then cut strip, measure the pieces, cut, measure the pieces, so on. This demonstrates that 'any' of the pairs of points will have a positive real number length. Obviously, this demonstration uses inductive rather than deductive reasoning.