Proving Theorems

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Proving Theorems

• Lesson 2.3
• Pre-AP Geometry

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Proofs

• Geometric proof is deductive reasoning at work.
• Throughout a deductive proof, the statements
  that are made are specific examples of more
  general situations, as is explained in the
  "reasons" column.
• Recall, a theorem is a statement that can be
  proved.

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Vocabulary

• Midpoint
• The point that divides, or bisects, a segment
  into two congruent segments.
• Bisect
• To divide into two congruent parts.
• Segment Bisector
• A segment, line, or plane that intersects a
  segment at its midpoint.

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Midpoint Theorem
If M is the midpoint of AB, then AM ½AB and MB
½AB
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Proof Midpoint Formula
Given M is the midpoint of Segment AB Prove
AM ½AB MB ½AB
Statement 1. M is the midpoints of segment
AB 2. Segment AM Segment MB, or AM MB 3.
AM MB AB 4. AM AM AB, or 2AM AB 5.
AM ½AB  6. MB ½AB
Reason 1. Given 2. Definition of midpoint 3.
Segment Addition Postulate 4. Substitution
Property (Steps 2 and 3) 5. Division
Prop. of  Equality 6. Substitution Property.
(Steps 2 and 5)
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The Midpoint Formula

• The Midpoint Formula
• If A(x1, y1) and B(x2, y2) are points in a
  coordinate plane, then the midpoint of segment AB
  has coordinates

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The Midpoint Formula

• Application
• Find the midpoint of the segment defined by the
  points A(5, 4) and B(-3, 2).

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Midpoint Formula

• Application
• Find the coordinates of the other endpoint B(x,
  y) of a segment with endpoint C(3, 0) and
  midpoint M(3, 4).

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Vocabulary

• Angle Bisector
• A ray that divides an angle into two adjacent
  angles that are congruent.

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Angle Bisector Theorem

• If BX is the bisector of ?ABC, then the measure
  of ? ABX is one half the measure of ?ABC and the
  measure of ?XBC one half of the ?ABC.

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Proof Angle Bisector Theorem

• Given BX is the bisector of ?ABC.
• Prove m ?ABX ½ m? ABC m XBC ½m? ABC

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Deductive Reasoning

• If we take a set of facts that are known or
  assumed to be true, deductive reasoning is a
  powerful way of extending that set of facts.
• In deductive reasoning, we say (argue) that if
  certain premises are known or assumed, a
  conclusion necessarily follows from these.
• Of course, deductive reasoning is not infallible
  the premises may not be true, or the line of
  reasoning itself may be wrong .

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Deductive Reasoning

• For example, if we are given the following
  premises
• A) All men are mortal,
• B) and Socrates is a man,
• then the conclusion Socrates is mortal
  follows from
• deductive reasoning.
• In this case, the deductive step is based on the
  logical principle that "if A implies B, and A is
  true, then B is true.

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Written Exercises

• Problem Set 2.3A, p. 46 1 12
• Problem Set 2.3B, P. 47 13 22
• Challenge p.48, Computer Key-In Project
  (optional)
• Submit a print out of your results from running
  the program along with your answers to Exercises
  1 3.
• Download BASIC at http//www.justbasic.com

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Computer Key-In Project