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2.1 Conditional 2.1 Conditional Statements Statements
2.1 Conditional Statements2.1 Conditional Statements
Goal 1: Recognizing Conditional Goal 1: Recognizing Conditional StatementStatement
A logical statement with 2 parts2 parts are called the hypothesis &
conclusionCan be written in “if-then” form; such as,
“If…, then…” Hypothesis is the part after the word “If”Conclusion is the part after the word
“then”
Ex: Underline the hypothesis & Ex: Underline the hypothesis & circle the conclusion.circle the conclusion.
If you are a brunette, then you have brown hair.
hypothesis conclusion
Ex: Rewrite the statement in “if-then” formEx: Rewrite the statement in “if-then” form
1. Vertical angles are congruent.
If there are 2 vertical angles, then they are congruent.
If 2 angles are vertical, then they are congruent.
Ex: Rewrite the statement in “if-then” formEx: Rewrite the statement in “if-then” form
2. An object weighs one ton if it weighs 2000 lbs.
If an object weighs 2000 lbs, then it weighs one ton.
CounterexampleCounterexample
Used to show a conditional statement is false.
It must keep the hypothesis true, but It must keep the hypothesis true, but the conclusion false!the conclusion false!
It must keep the hypothesis true, but It must keep the hypothesis true, but the conclusion false!the conclusion false!
It must keep the hypothesis true, but It must keep the hypothesis true, but the conclusion false!the conclusion false!
Ex: Find a counterexample to Ex: Find a counterexample to prove the statement is false.prove the statement is false.
If x2=81, then x must equal 9.
counterexample: x could be -9
because (-9)2=81, but x≠9.
NegationNegation
Writing the opposite of a statement.
Ex: negate x=3
x≠3Ex: negate t>5
t 5
ConverseConverse
Switch the hypothesis & conclusion parts of a conditional statement.
Ex: Write the converse of “If you are a brunette, then you have brown hair.”
If you have brown hair, then you are a brunette.
InverseInverse
Negate the hypothesis & conclusion of a conditional statement.
Ex: Write the inverse of “If you are a brunette, then you have brown hair.”
If you are not a brunette, then you do not have brown hair.
ContrapositiveContrapositive
Negate, then switch the hypothesis & conclusion of a conditional statement (negate the converse)
Ex: Write the contrapositive of “If you are a brunette, then you have brown hair.”
If you do not have brown hair, then you are not a brunette.
Equivalent Statements – both are true or both are false.
The original conditional statement & its contrapositive will always have the same meaning.
The converse & inverse of a conditional statement will always have the same meaning.
Equivalent StatementsEquivalent Statements
Goal 2: Using Point, Line, and Goal 2: Using Point, Line, and Plane PostulatesPlane Postulates
Postulate 5: Through any 2 points, there exists exactly one line.Postulate 6: A line contains at least 2 points.Postulate 7: If two lines intersect, then their intersection is exactly one point.Postulate 8: Through any 3 noncollinear points, there exists exactly one plane.Postulate 9: A plane contains at least 3 noncollinear points.Postulate 10: If 2 points lie in a plane, then the line containing them lies in the plane.Postulate 11: If 2 planes intersect, then their intersection is a line.
Identifying PostulatesIdentifying PostulatesThere is exactly one line (line n) that passes through the points A and B.
Line n contains at least two points, A and B.
Lines m and n intersect at point A.
Plane P passes through the noncollinear points A, B, and C.
Plane P contains at least three noncollinear points, A, B, and C.
Points A and B lie in plane P. So, line n, which contains points A and B, also lies in plane P.
Planes P and Q intersect. So, they intersect in a line, labeled in the diagram as line m.
Postulate 5
Postulate 6
Postulate 7
Postulate 8
Postulate 9
Postulate 10
Postulate 11
