Download Recognizing Conditional Statements

2.1 Conditional Statements
Goals
•Recognize a conditional statement
•Write postulates about points, lines
and planes
Recognizing Conditional Statements
Conditional Statements
If-Then Statements
If a number is divisible by both 2 and 3 then it is divisible by 6.
HYPOTHESIS
CONCLUSION
If a polygon has four sides then it is a quadrilateral.
If a number greater than two is even, then it is not prime.
I will dry the dishes if you wash them.
Recognizing Conditional Statements
Conditional statements can be True or
False
• To show a conditional statement is true, you
must present an argument to show true in all
cases.
• To show conditional statement is false, you
only have to have a single counterexample.
Recognizing Conditional Statements
Example:
Write a counterexample:
If a number is odd, then it is divisible by 3
Recognizing Conditional Statements
Example 1
State the hypothesis and
conclusion for each statement.
IF two angles are supplementary, THEN the
sum of their angles is 180 degrees.
IF you are 5 feet tall, THEN are also 60 inches
tall.
Recognizing Conditional Statements
Example 1
State the hypothesis and
conclusion for each statement.
IF two angles are adjacent, THEN they have
a common vertex.
Three noncollinear points are coplanar IF
they lie on the same plane.
Recognizing Conditional Statements
Example 2
Rewrite in if-then form
All monkeys have tails.
Vertical angles are congruent.
Recognizing Conditional Statements
Example 2
Rewrite in if-then form
Supplementary angles have measures
whose sum is 180°.
Practice is cancelled if it rains.
Recognizing Conditional Statements
The CONVERSE of a conditional statement
is formed by interchanging the hypothesis
and conclusion.
conditional statement
If x – y is positive then x > y .
converse
If x > y then x – y is positive.
Recognizing Conditional Statements
If the CONVERSE statement is true the
converse may or may not be true.
conditional statement
If two angles are adjacent they share
a common side.
Recognizing Conditional Statements
1. IF two angles are adjacent, THEN they have a
common vertex.
CONVERSE - IF two angles have a common vertex,
THEN they are adjacent.
2. IF two angles are supplementary, THEN the sum
of their angles is 180 degrees.
CONVERSE - IF two angles have a sum of 180
degrees, THEN they are supplementary.
3. IF you are 5 feet tall, THEN are also 60 inches tall.
CONVERSE - IF you are 60 inches tall, THEN are
also 5 feet tall.
Recognizing Conditional Statements
The denial of a statement is called a NEGATION.
RST is an obtuse angle.
Intersecting lines are coplanar.
If we take a test today we do not have
homework.
Recognizing Conditional Statements
Given a conditional statement, its INVERSE can
be formed by negating both the hypothesis and
conclusion.
The inverse of a true statement is not necessarily true.
EXAMPLE
Conditional statement: If the angle is 75 degrees, then it
is acute.
Inverse: If the angle is not 75 degrees, then it is not
acute.
Recognizing Conditional Statements
Example 3
Find the inverse of the following statement. Is it True
or False
If you have vertical angles, then they
are congruent.
Recognizing Conditional Statements
CONTRAPOSITIVE: Formed by negating the
hypothesis and conclusion of the converse of
the given conditional.
When forming a contrapositive of a conditional it may be
easier to write the converse first – then negate each part.
Example:
Statement: If the angle is 75 degrees then it is acute .
Recognizing Conditional Statements
When two statements are both true or both false, they
are called equivalent statements.
Original
If mA = 30°, then  A is acute.
Inverse
If mA  30°, then  A is not acute.
Converse
If  A is acute then mA = 30°.
F
T
Contrapositive If  A is not acute then mA  30°.
A conditional statement is equivalent to its
contrapositive.
The inverse and converse of any conditional statement
are equivalent.
Recognizing Conditional Statements
Example 5:
Write the contrapositive of the conditional statement
If two angles are vertical, then they are congruent.
Using Point, Line and Plane Postulates
Postulate 5 Two Points - Line
Through any two points
there is exactly one line.
(as an If-then statement)
If there are two points, then there is exactly
one line that contains them.
Using Point, Line and Plane Postulates
Postulate 6
Line - Two Points
A line contains at
least two points.
Using Point, Line and Plane Postulates
Postulate 7 Three Points - Plane
If two lines intersect, then their
intersection is exactly one point.
Using Point, Line and Plane Postulates
Postulate 8 Three Points - Plane
Through any three non
collinear points there is
exactly one plane.
(or as an If-then statement)
If there are three non collinear points, then there is exactly
one plane that contains them.
Using Point, Line and Plane Postulates
Postulate 9 Plane - Three Points
A plane contains
at least three non
collinear points.
Using Point, Line and Plane Postulates
Postulate 10 Two Points - Line - Plane
If two points lie in a plane, then
the entire line containing those
two points lies in the plane.
Using Point, Line and Plane Postulates
Postulate 11 Plane Intersection - Line
If two planes
intersect, then their
intersection is a
line.
Using Point, Line and Plane Postulates
Example:
Decide whether the statement is True or False.
If False, give a counterexample.
Three points are always contained in a line.
Homework
2.1 10-46 mult of 3