2'5 Postulates and Paragraph Proofs 101308

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2.5 Postulates and Paragraph Proofs 10/13/08

• In geometry, a postulate, or an axiom, is a
  statement that describes a fundamental
  relationship between the basic terms of geometry.
• Postulates are accepted as true.
• The basic ideas about points, lines, and planes
  can be stated in postulates.

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Postulates

• 2.1 Through any two points, there is exactly one
  line.
• 2.2 Through any three points not on the same
  line, there is exactly one plane.
• 2.3 A line contains at least two points.
• 2.4 A plane contains at least three points not on
  the same line.
• 2.5 If two points lie in a plane, then the entire
  line containing those points lies in that plane.
• 2.6 If two lines intersect, then their
  intersection is exactly one point.
• 2.7 If two planes intersect, then their
  intersection is a line.

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Use Postulates

• Determine whether each statement is always,
  sometimes, or never true.
• If points A, B, and C lie in plane M, then they
  are collinear.
• Sometimes A, B, and C do not necessarily have
  to be collinear to lie in a plane.
• There is exactly one plane that contains
  noncollinear points P, Q, and R.
• Always Postulate 2.2 states that through any
  three noncollinear points, there is exactly one
  plane.
• There are at least two lines through points M and
  N.
• Never postulate 2.1 states that through any
  two points, there is exactly one line.

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Paragraph Proofs

• Once a statement or conjecture has been shown to
  be true, it is called a theorem.
• A proof is a logical argument in which each
  statement you make is supported by a statement
  that is accepted as true.
• One type of proof is called a paragraph proof, or
  informal proof.
• In this type of proof, you write a paragraph to
  explain why a conjecture for a given statement is
  true.

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5 Essential Parts of a Good Proof

• State the theorem or conjecture to be proven.
• List the given information.
• If possible, draw a diagram to illustrate the
  given information.
• State what is to be proved.
• Develop a system of deductive reasoning.

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Write a Paragraph Proof

• Given that M is the midpoint of , write a
  paragraph proof to show that
• Given M is the midpoint of
• Prove
• From the definition of midpoint of a segment, PM
  MQ. This shows that the segments have the
  same measure. By the definition of congruence,
  if two segments have the same measure, then they
  are congruent. So,

Q
M
P
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Midpoint Theorem

• If M is the midpoint of

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Homework!!!

• PSAT math sections