Transcript Conditional Statements

Conditional Statements
• Logical statements that have two parts, a
hypothesis and a conclusion.
• The “if” part contains the hypothesis and
the “then” part the conclusion.
• Ex: If it is raining then there are clouds in
the sky.
• Make the following statement a conditional
statement: Two angles are supplementary
if they are a linear pair.
Conditional Statements
• The negation of a statement is the
opposite of the original. Ex: We are in
Geometry class. We are not in Geometry
class.
• Conditional Statements may be true or
false. They needed to be proven one way
or the other. You must show examples of
it being true or use a counterexample to
prove it false.
Conditional Statements
• The converse of a conditional statement
switches the hypothesis with the
conclusion.
• The inverse negates both the hypothesis
and the conclusion.
• The contrapositive is the converse with
both the conclusion and hypothesis
negated.
Examples
• Conditional- If angle A=99 degrees, then
angle A is obtuse.
• Converse- If angle A is obtuse, then angle
A=99 degrees.
• Inverse- If angle A is not equal to 99
degrees, then angle A is not obtuse.
• Contrapositive- If angle A is not obtuse,
then angle A is not equal to 99 degrees.
• Which are true and which are false?
Equivalent Statements
• A conditional statement and its
contrapositive are either both true or both
false.
• The converse and inverse of the
conditional are either both true or both
false.
• These are examples of equivalent
statements.
Definitions and Perpendicular Lines
• Definitions can be written as conditional
statements or as the converse of the
statement if both are true.
• The definition of perpendicular lines can
be written in either of these forms.
• If two lines intersect to form a right angle,
then they are perpendicular.
• If two lines are perpendicular, then they
intersect to form a right angle.
Biconditional Statements
• When both the conditional statement and
its converse are true, you can write them
as a single biconditional statement using
“if and only if.”
• Any valid definition can be written as a
biconditional statement.
• Two lines are perpendicular if and only if
they intersect to form a right angle.
Postulates
• Postulate 5 – Through any two points
there exists exactly one line.
• Postulate 6 – A line contains at least two
points.
• Postulate 7 – If two lines intersect, then
their intersection is exactly one point.
• Postulate 8 – Through any three
noncollinear points there exists exactly
one plane.
Postulates
• Postulate 9 – A plane contains at least
three noncollinear points.
• Postulate 10 – If two points lie in a plane,
then the line containing them lies in the
plane.
• Postulate 11 – If two planes intersect, then
their intersection is a line.
Properties
• Properties of Equality: Addition,
Subtraction, Multiplication and Division.
• Distributive and Substitution.
• Reflexive
• Symmetric
• Transitive
Angle Pair Relationship Theorems
• Congruent Supplements
• Congruent Complements
• Right Angles Congruence
Given: Angles 1 and 2 and angles 3 and 2 are
supplements.
Prove: Angle 1 is congruent to angle 3.
• Angles 1 and 2 are
supplements
• Angles 3 and 2 are
supplements
• M1 + M2 = 180
• M3 + M2 = 180
• M1+M2=M3+M2
• M1=M3
• Angle 1 is congruent
to angle 3
Given: Angles 1 and 2 and angles 3 and 2 are
supplements.
Prove: Angle 1 is congruent to angle 3.
• Angles 1 and 2 are
supplements
• Angles 3 and 2 are
supplements
• M1 + M2 = 180
• M3 + M2 = 180
• M1+M2=M3+M2
• M1=M3
• Angle 1 is congruent to
angle 3
• Given
• Given
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Def. of Supp. Angles
Def. of Supp. Angles
Transitive Property
Subtraction Property
Def of Congruent angles
Given: Angle’s 1&2 are a linear pair
Angles 1&2 are congruent
Prove: g perpendicular h
g
1
2
h
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1 and 2 are a linear pair
1 and 2 are supplementary
1 + 2 = 180
1 congruent 2
M1 = m2
M1 + m1 = 180
2(m1) =180
M1=90
1 is a right angle
G perpendicular to h
Given: Angle’s 1&2 are a linear pair
Angles 1&2 are congruent
Prove: g perpendicular h
g
1
2
g
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1 and 2 are a linear pair
1 and 2 are supplementary
1 + 2 = 180
1 congruent 2
M1 = m2
M1 = m1 = 180
2(m1) =180
M1=90
1 is a right angle
G perpendicular to h
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Given
Linear pair postulate
Def. of supp. Angles
Given
Def. of congruent angles
Substitution Prop. Of Equality
Combine like Terms
Division Prop. Of Equality
Def. of right angle
Def. of Perpendicular lines