• Conditional: If the car is running, then it has fuel
• 1) Write the converse
• 2) Write the “opposite” statement of the conditional
• 3) Write the “opposite” statement of the converse

2.0 Students write geometric proofs, including proofs by contradiction

• Conditional
– If p , then q
Conditional:
– If it snows tomorrow , then we will go skiing
• Converse
– If q , then p
Converse:
– If we go skiing , then it snows tomorrow

• Biconditional
– When both the conditional and converse are both true statements
– Joins the hypothesis and conclusion with if and only if
– Example: A point is on the perpendicular bisector of a segment if and only if it is equidistant from the endpoints of the segment

• Negation has the opposite meaning of the original statement
• Example:
Statement: “An angle is a straight angle”
Negation: “An angle is NOT a straight angle”

• Write the negation of a statement
• Statement : “Angle ABC is obtuse”
• Negation :
• “Angle ABC is NOT obtuse”

• Write the negation of the statement
• Statement : “Lines m and n are not perpendicular”
• Negation :
• “Line m and n ARE perpendicular”

• Inverse : negates both the hypothesis and the conclusion
– If ~p , then ~q (if not p, then not q)
Conditional:
– If it snows tomorrow , then we will go skiing
Inverse:
– If it does not snow tomorrow , then we will not go skiing

• Write the inverse of the conditional statement
• Conditional : “If a figure is a square, then it is a rectangle”
• Inverse :
• “If a figure is NOT a square, then it is NOT a rectangle”

• Write an inverse for the conditional statement
• Conditional : “If two angles add up to 180, then they are supplementary”
• Inverse :
• “If two angles do not add up to 180, then they are not supplementary”

• Contrapositive : Switches the hypothesis and the conclusion and negates both
– If not q , then not p
– Conditional: “If it snows tomorrow , then we will go skiing”
– Contrapositive:
– “If we do not go skiing , then it does not snows tomorrow”

• Find the Contrapositives
• Conditional Statement : “If an angle is a straight angle, then its measure is 180”
• Contrapositives :
• “If an angle’s measure is not 180, then it is not a straight angle”

• Find the contrapositive
• Conditional : “If two lines are parallel, then they do not intersect”
• Contrapositive :
• “If they do intersect, then two lines are not parallel”

Statement : The angle is obtuse
Negation :
The angle is not obtuse

• Equivalent statements = statements with the same truth value
• Example:
– “If a figure is a square, then it is a rectangle”
(Conditional)
– “If a figure is not a rectangle, then it is not a square” (Contrapositive)
– These two statements are the same

• Indirect reasoning = type of reasoning in which all possibilities are considered and then the unwanted ones are proved false. The remaining possibilities must be true.

• Indirect proof = see indirect reasoning

• Write the negation
– Statement: “Today is not Tuesday”
– Negation:

• Step 1) State as an assumption the opposite
(negation) of what you want to prove
• Step 2) Show that this assumption leads to a contradiction
• Step 3) Conclude that the assumption must be false and that what you want to prove must be true

• If Jaeleen spends more than $50 to buy two items at a clothing store , then at least one of the items costs more than $25 Dollars
– Given: The cost of two items is more than $50
– Prove: At least one of the items costs more than $25
• Step 1) Assume negation of what you are trying to prove is true. “Neither item costs more than $25”
• Step 2) This means that each item costs $25 or less, which lead to that two items together cost $50 or less.
Which contradict the given
• Step 3) So the negation is false. So one item must cost more than $25

• Identify the two statements that contradict each other
– I. ABC is acute
– II. ABC is scalene
– III. ABC is equiangular

• Pgs 283-284 #2-20 even, 21