Unit 2 Logic and Proof

1
Unit 2Logic and Proof

• Learn deductive logic
• Do your first 2-column proof
• New Theorems and Postulates

2
Lecture 1 (2-1)

• Objectives
• Recognize the hypothesis and conclusion of an
  if-then statement
• State the converse of an if-then statement
• Use a counterexample
• Understand if and only if

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The If-Then Statement

• Conditional

• is a two part statement with an actual or
  implied if-then.

If p, then q.
p ---gt q
hypothesis
conclusion
If the sun is shining, then it is daytime.
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Hidden If-Thens
A conditional may not contain either if or then!
Two intersecting lines are contained in exactly
one plane.
Which is the hypothesis? Which is the conclusion?
two lines intersect
exactly one plane contains them
The whole thing
If two lines intersect, then exactly one plane
contains them.
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The Converse

• A conditional with the hypothesis and conclusion
  reversed.

Original If the sun is shining, then it is
daytime.
If q, then p.
q ---gt p
hypothesis
conclusion
If it is daytime, then the sun is shining.
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The Counterexample

• The only way a conditional can be false is if the
  hypothesis is true and the conclusion is false.
  This is called a counterexample.

If x gt 5, then x 6. If x 5, then 4x 20
x could be equal to 5.5 or 7 etc always true,
no counterexample
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The Biconditional

• If a conditional and its converse are the same
  (same truth) then it is a biconditional and can
  use the if and only if language.

If m?1 90?, then ?1 is a right angle. m?1
90? if and only if ?1 is a right angle.
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Lecture 2 (2-2)

• Objectives
• Do your first proof
• Use the properties of algebra and the properties
  of congruence in proofs

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Properties from Algebra

• see properties on page 37

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Your First Proof
Given 3x 7 - 8x 22
Prove x - 3
STATEMENTS REASONS

• 1. 3x 7 - 8x 22 1. Given
• 2. -5x 7 22 2.
  Substitution
• 3. -5x 15 3.
  Subtraction Prop.
• 4. x - 3 4.
  Division Prop.

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Your Second Proof
A
B
Given AB CD
C
D
Prove AC BD
STATEMENTS REASONS

• 1. AB CD 1. Given
• 2. AB BC BC CD 2. Addition Prop.
• 3. AB BC AC 3. SAP
• BC CD BD
• 4. AC BD 4.
  Substitution

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Lecture 3 (2-3)

• Objectives
• Use the Midpoint Theorem and the Bisector Theorem
• Know the kinds of reasons that can be used in
  proofs

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The Midpoint Theorem

• If M is the midpoint of AB, then
• AM ½ AB and MB ½ AB

B
A
M
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The Bisector Theorem

• If BX is the bisector of ?ABC, then
• m ? ABX ½ m ? ABC
• m ? XBC ½ m ? ABC

A
X
B
C
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Reasons Used in Proofs

• Given Information
• Definitions
• Postulates (including Algebra)
• Theorems

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Lecture 4 (2-4)

• Objectives
• Apply the definitions of complimentary and
  supplementary angles
• State and apply the theorem about vertical angles

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Complimentary Angles

• Any two angles whose measures sum to 90.
• If m?ABC m? SXT 90, then
• ? ABC and ? SXT are complimentary.

S
A
X
C
B
T
See It!
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Supplementary Angles

• Any two angles whose measures sum to 180.
• If m?ABC m? SXT 180, then
• ? ABC and ? SXT are supplementary.

S
A
X
C
T
B
See It!
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Vertical Angles

• Two angles formed on the opposite sides of the
  intersection of two lines.

1
2
4
3
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Vertical Angle Theorem

• Vertical angles are congruent

1
2
4
3
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Lecture 5 (2-5)

• Objectives
• Recognize perpendicular lines
• Use the theorems about perpendicular lines

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Perpendicular Lines

• Two lines that intersect to form right angles.

If l ? m, then angles are right.
l
m
See It!
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Theorem 2-4

• If two lines are perpendicular, then they form
  congruent, adjacent angles.

l
If l ? m, then ?1 ? ? 2.
2
1
m
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Theorem 2-5

• If two lines intersect to form congruent,
  adjacent angles, then the lines are perpendicular.

l
If ?1 ? ? 2, then l ? m.
2
1
m
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Theorem 2-6

• If the exterior sides of two adjacent angles lie
  on perpendicular lines, then the angles are
  complimentary.

l
If l ? m, then ?1 and ? 2 are compl.
1
2
m
See It!
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Lecture 6 (2-6)

• Objectives
• Discover the steps used to plan a proof

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Theorem 2-7

• If two angles are supplementary to congruent
  angles (the same angle) then they are congruent.

If ?1 suppl ? 2 and ? 2 suppl ? 3, then ? 1 ? ?
3.
1
2
3
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Theorem 2-8

• If two angles are complimentary to congruent
  angles (or to the same angle) then they are
  congruent.

If ?1 compl ? 2 and ? 2 compl ? 3, then ? 1 ? ?
3.
1
2
3
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Planning a Proof

• Mark the given on the figure
• See the connection
• Jot some notes of things that might be useful
• Complete the proof