4.1 MTH 97 Rogers notes.notebook
October 21, 2010
Section 4.1 Reasoning and Proof
Direct or Deductive Reasoning: draw a conclusion from a series of
statements (often represented by letters p and q).
Conditional Statements: "if p, then q" format. (Written p
Hypothesis is "p" and Conclusion is "q."
q)
Example: What can we conclude from the following statements?
1. If a point is equidistant from the endpoints of a segment, then
the point lies on the perpendicular bisector of the segment.
2. Point P is equidistant from points A and B.
Conclusion:
(three-dot triangle means "therefore" (see pg 187)
Law of Detachment:"q" becomes detached from p
know "p."
q once we
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4.1 MTH 97 Rogers notes.notebook
October 21, 2010
Law of Syllogism: relates several "if..., then" statements:
If p, then q.
If q, then r.
Therefore if p, then r
Symbolically:
Example:
If a quadrilateral has four congruent sides,
then it is a rhombus.
If a quadrilateral is a rhombus,
then its diagonals are perpendicular.
(given:) A quadrilateral has four congruent sides,
Therefore:
Symbolically:
(look at Example 4.3 on pp. 187-188)
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4.1 MTH 97 Rogers notes.notebook
October 21, 2010
Example:
If XYZW is a square,
then it has four right angles.
If a quadrilateral has 4 rt. angles,
then it is a rectangle.
If a quadrilateral is a rectangle,
then its opposite sides are parallel.
If a quadrilateral has opposite sides parallel,
then it is a parallelogram.
(given:) XYZW is a square.
Therefore:
Symbolically:
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4.1 MTH 97 Rogers notes.notebook
October 21, 2010
Conditional: "If p..., then q" p q
Converse: "If q..., then p" q p
Inverse: "If not p..., then not q" not p not q
Contrapositive: "If not q..., then not p" not q not p
If both the Conditional statement and its Converse are true we can make the Biconditional Statement: "p if and only if q" p q
Example:
Conditional:
If a square has an area of 25 cm2,
then the length of its side is 5 cm. p q True or False?
Converse:
If the length of the side of a square is 5 cm,
then it has an area of 25 cm2. q p True or False?
Biconditional:
A square has area 25 cm2 if and only if the length of a side is 5 cm.
p q
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4.1 MTH 97 Rogers notes.notebook
October 21, 2010
Read together the Vertical Angles Proofs pp. 190-191
Paragraph proof vs. Statement/Reasons Column proof:
You try to complete the proof on page 194 problem #45.
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