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2.5 – Postulates and Paragraph Proofs

2.5 – Postulates and Paragraph Proofs. Postulates and Axioms. Statements that are accepted as true without proof. Basic ideas about points, lines and planes can be stated as postulates. Postulates. See Table, pg. 127 on Textbook. Post 2.1 – Through any two points there is exactly one line.

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2.5 – Postulates and Paragraph Proofs

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  1. 2.5 – Postulates and Paragraph Proofs

  2. Postulates and Axioms • Statements that are accepted as true without proof. • Basic ideas about points, lines and planes can be stated as postulates.

  3. Postulates • See Table, pg. 127 on Textbook. Post 2.1 – Through any two points there is exactly one line. Post 2.2 – Though any three non collinear points there is exactly one plane.

  4. Postulates • See Table, pg. 127 on Textbook. Post 2.3 – A line contains at least two points. Post 2.4 – A plane contains at least three non collinear points. Post 2.5 – If two points lie on a plane, then the line containing those points also lies in that plane.

  5. Postulates • See Table, pg. 127 on Textbook. Post 2.6 – Two lines intersect to form exactly one point. Post 2.7 – Two planes intersect to form exactly one line.

  6. Postulates • Example: • Determine whether the following are Always, Sometimes or Never true. Explain your reasoning. • If two coplanar lines intersect, their intersection point also lies on the same plane.

  7. Postulates • Example: • Determine whether the following are Always, Sometimes or Never true. Explain your reasoning. • Any four points are non collinear.

  8. Postulates • Example: • Determine whether the following are Always, Sometimes or Never true. Explain your reasoning. • Line GH contains three non collinear points.

  9. Paragraph Proofs • To prove a conjecture, use deductive reasoning to move from Hypothesis to Conclusion. • This is done by writing a “proof.” • Proof – logical argument in which statements you make must be supported by a Postulate/Axiom.

  10. Paragraph Proofs • Once a statement or conjecture has been proven, it is called a Theorem. • Theorems can be used to justify arguments in proofs as well. • A paragraph proof involves writing a paragraph to show that a conjecture is true.

  11. Paragraph Proofs • Paragraph proofs are referred to as “informal proofs.” • The Mid Point Theorem is an example of a conjecture that has been proven true and is now used to justify other arguments.

  12. Midpoint Theorem • If M is the midpoint of segment AB, then segment AM is congruent to segment MB. A M B

  13. Paragraph Proof Example • Given Point M is the Midpoint of Segment XY, Prove that XM and MY are congruent. • The midpoint of a segment divides that segment into two segments of the same length. Therefore, XM and MY have the same length. Segments that have the same length are congruent. Therefore, XM and MY are congruent.`

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