UNIT 2 – DAY 4 CONDITIONAL STATEMENTS CONVERSES BICONDITIONALS Honors Geometry
Feb 15, 2016
UNIT 2 – DAY 4
CONDITIONAL STATEMENTSCONVERSES
BICONDITIONALS
Honors Geometry
Warm Up
What is the fourth point of plane XUR
Name the intersection of planes QUV and QTX
Are point U and S collinear?
Quiz
2-1 Conditional Statements
Objectives To recognize conditional statements To write converses of conditional statements
If-Then Statements
“If it is Valentine’s Day, then it is February.”
Another name of an if-then statement is a conditional. Parts of a Conditional:
Hypothesis (after “If”) Conclusion (after “Then”)
“If you are not completely satisfied, then your money will be refunded.”
(hypothesis) (conclusion)
Identifying the Parts
Identify the hypothesis and the conclusion of this conditional statement:
If it is Halloween, then it is October
Hypothesis: It is Halloween Conclusion: It is October
Writing a Conditional
Write each sentence as a conditional: A rectangle has four right angles
“If a figure is a rectangle, then it has four right angles.”
An integer that ends with 0 is divisible by 5
“If an integer ends with 0, then it is divisible by 5.”
Truth Value
A conditional can have a truth value of true or false.
To show that a conditional is true, you must show that every time the hypothesis is true, the conclusion is also true.
To show that a conditional is false, you need to only find one counterexample
Example
Show that this conditional is false by finding a counterexample “If it is February, then there are only 28 days in the
month”
Finding one counterexample will show that this conditional is false
February 2012 is a counterexample because 2012 was a leap year and there were 29 days in February
Converses
The converse of a conditional switches the hypothesis and the conclusion
Example Conditional: “If two lines intersect to form right
angles, then they are perpendicular.”
Converse: “If two lines are perpendicular, then they intersect to form right angles.”
Example
Write the converse of the following conditional:
“If two lines are not parallel and do not intersect, then they are skew”
“If two lines are skew, then they are not parallel and do not intersect.”
Are all converses true?
Write the converse of the following true conditional statement. Then, determine its truth value. Conditional: “If a figure is a square, then it has four
sides”
Converse: “If a figure has four sides, then it is a square”
Is the converse true?
NO! A rectangle that is not a square is a counterexample!
Assessment Prompt
Write the converse of each conditional statement. Determine the truth value of the conditional and its converse.1. If two lines do not intersect, then they are parallel
Converse: “If two lines are parallel, then they do not intersect.”
Conditional is false Converse is true
2. If x = 2, then |x| = 2 Converse: “If |x| = 2, then x = 2”
Conditional is true Converse if false
2-2 Biconditionals
Objectives To write biconditionals
2-2 Biconditionals
When a conditional and its converse are true, you can combine them as a biconditional.
This is a statement you get by connecting the conditional and its converse with the phrase if and only if (iff).
Example of a Biconditional
Conditional If two angles have the same measure, then the angles
are congruent. True
Converse If two angles are congruent, then the angles have the
same measure. True
Biconditional Two angles have the same measure if and only if the
angles are congruent.
Example
Consider this true conditional statement. Write its converse. If the converse is also true, combine them as a biconditional If three points are collinear, then they lie on the same
line.
If three points lie on the same line, then they are collinear.
Three points are collinear if and only if they lie on the same line.
Definitions
A good definition is a statement that can help you identify or classify an object.
A good definition can be written as a biconditional.
Example
Show that this definition of perpendicular lines is a good defintion and that it can be written as a biconditional Definition: Perpendicular lines are two lines that
intersect to form right angles. Conditional: If two lines are perpendicular, then they
intersect to form right angles. Converse: If two lines intersect to form right angles,
then they are perpendicular. Biconditional: Two lines are perpendicular if and only
if they intersect to form right angles.
Real World Examples
Are the following statements good definitions? Explain An airplane is a vehicle that flies.
Can it be written as a biconditional? NO! A helicopter is a counterexample because it also
flies!
A triangle has sharp corners. Can it be written as a biconditional? NO! Squares have sharp corners. (Sharp is not a precise
word)
Homework
Worksheet
Logic Quiz Monday!