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2-5 Postulates and Paragraph Proofs

Learn about postulates and paragraph proofs in geometry, including examples and exercises to deepen your understanding. Discover the essential parts of a good proof and how to apply deductive reasoning.

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2-5 Postulates and Paragraph Proofs

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  1. 2-5 Postulates and Paragraph Proofs • Postulate or Axiom: A statement that is accepted as true • Postulate 2.1: Through any two points, there is exactly one line. • Postulate 2.2: Through any three points not on the same line, there is exactly one plane.

  2. Example #1 • Review Example #1 on page 105 and then complete the following example… • Some snow crystals are shaped like regular hexagons. How many lines must be drawn to interconnect all vertices of a hexagonal snow crystal? • Answer: 15

  3. More Postulates • 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. • Postulate 2.5: 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. Example #2 • Determine whether each statement is always, sometimes, or never true. Explain. • 1) If points A, B, and C lie in plane M, then they are collinear. • Sometimes • 2) There are at least two lines through points M and N. • Never • 3) If two coplanar lines intersect, then the point of intersection lies in the same plane as the two lines. • Always • 4) GH contains three noncollinear points. • Never

  5. Paragraph Proofs • Theorem: A statement or conjecture that has been shown or proven to be true. • Proof: A logical argument in which each statement you make is supported by a statement that is accepted as true. • Paragraph Proof (informal proof): One type of proof.

  6. 5 essential parts of a good proof • 1) State the theorem or conjecture to be proven • 2) List the given information • 3) If possible, draw a diagram to illustrate the given information • 4) State what is to be proved • 5) Develop a system of deductive reasoning

  7. C B A Example #3 • Given that , and C is between A and B, write a paragraph proof to show that C is the midpoint of AB. Midpoint Theorem If M is the midpoint of AB, then

  8. Homework #13 • p. 108 8-21, 28-29 • Quiz Monday

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