2.3. Converse, Inverse, and Contrapositive http://www.ck12.org
Notice for the inverse and conversewe can use the same counterexample.This is because the inverse and converse
are alsologically equivalent.
Example C
The following is a true statement:
m^6 ABC> 90 ◦if and only if^6 ABCis an obtuse angle.
Determine the two true statements within this biconditional.
Statement1: Ifm^6 ABC> 90 ◦, then^6 ABCis an obtuse angle
Statement2: If^6 ABCis an obtuse angle, thenm^6 ABC> 90 ◦.
You should recognize this as the definition of an obtuse angle. All geometric definitions are biconditional statements.
Watch this video for help with the Examples above.
MEDIA
Click image to the left for use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/52466CK-12 Foundation: Chapter2ConverseInverseContrapositiveB
Concept Problem Revisited
Your sister presented you with the if-then statement, "If you do the dishes, then I will help you with your homework."
If we take the original statement to be true, then the contrapositive is also true. The following contrapositive
statement islogically equivalentto the original if-then statement:
"If I do not help you with your homework, then you will not do the dishes."
Vocabulary
Aconditional statement(also called anif-then statement) is a statement with a hypothesis followed by a conclusion.
Thehypothesisis the first, or “if,” part of a conditional statement. Theconclusionis the second, or “then,” part of a
conditional statement. The conclusion is the result of a hypothesis. Theconverseof a conditional statement is when
the hypothesis and conclusion are switched. Theinverseof a conditional statement is when both the hypothesis and
conclusions are negated. Thecontrapositiveof a conditional statement is when the hypothesis and conclusions have
been both switched and negated. When the original statement and converse are both true then the statement is a
biconditional statement.
Guided Practice
- Use the statement: Any two points are collinear.
a) Find the converse, inverse, and contrapositive.
b) Determine if the statements from part a are true or false. If they are false, find a counterexample.
2.p:x< 10 q: 2x< 50