Transcript Recognizing Conditional Statements

Conditional Statements
Goals
•Recognize a conditional statement
•Write the converse, inverse, and
conditional statement
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.
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.
Hypothesis
Conclusion
IF two angles are supplementary, THEN the
sum of their angles is 180 degrees.
Recognizing Conditional Statements
Example 2
State the hypothesis and
conclusion for each statement.
IF two angles are adjacent, THEN they have
a common vertex.
Hypothesis: Two angles are adjacent
Conclusion: The angles have a common
vertex
Recognizing Conditional Statements
Example 2
Rewrite in if-then form
All monkeys have tails.
If an animal is a monkey, then the animal has a
tail.
Vertical angles are congruent.
If two angles are vertical, then they are
congruent.
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
Write the converse of the following 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.
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.
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.
If angles are not vertical angles, then they are not
congruent.
False
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 .
If an angle is not acute then it is not 75 degrees
Recognizing Conditional Statements
Example 5:
Write the contrapositive of the conditional statement
If two angles are vertical, then they are congruent.
If two angles are not congruent, then they are not vertical
angles
Biconditional Statements
• Biconditional statement is when the conditional
statement and converse are both true. It can be
written as an “if and only if” statement.
Conditional Statement:
If an angle is classified as a right angle, then it
measures 90 degrees
Is the conditional and converse True? If so, then…
• An angle is called a right angle if and only if it
measures 90 degrees.
Recognizing Conditional 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°.
Contrapositive If  A is not acute then mA  30°.
Write the converse, inverse, contrapositive, and
biconditional statement for the following conditional
statement.
If a triangle is isosceles, then it has two
congruent sides.