Download 2.3 Biconditionals and Definitions

Bellringer
• Find a counterexample to show that each
statement is false.
– 1. You can connect any four points to form a
rectangle.
– 2. The square of a number is always greater than the
number.
• Write the converse of each statement ;
determine the truth value.
– 3. If today is September, then tomorrow is October 1.
– 4. If AB is the perpendicular bisector of CD, then AB
and CD are perpendicular.
Answers
• 1. 4 collinear points
• 2. 0.5
• 3. If tomorrow is Oct. 1, then today is Sept; both
• 4. If AB and CD are perpendicular, then AB is the
perpendicular bisector of CD
Geometry
Chapter 2: Reasoning and Proofs
2.3 Biconditionals and
Definitions
Revisiting 2.2 Conditionals
• Conditional– Original statement
• Consists of hypothesis and conclusion
• If and then statement : p→ q
• Converse– Statement switches p and q
– Becomes conclusion then hypothesis
– If and then statement is q → p
Q=then you
should eat
vegetables.
P=If you
want to be
healthy
2.2 Conditional con’t
• Inverse– Original statement with Negation (meaning not) both
the hypothesis and conclusion
• If not p then not q : ~p→~q
• Contrapositive– Converse statement with Negation both the
conclusion and hypothesis.
• If not q then not p: ~q → ~p
Statement
Hypothesis
Conclusion
Conditional
If
P
Then
Q
Converse
If
Q
Then
P
Inverse
If not
P
Then not
Q
Contrapostive
If not
Q
Then not
P
Lesson Purpose
Objective
Essential Question
• To write biconditionals
and recognize good
definitions.
• How can you make a
conjecture and prove that
it’s true?
Definitions
• A bi-conditional – is a single true statement
• that combines a true conditional and its converse.
• Combines p→q and q→p as p ↔ q or q ↔ p.
• “p if and only if q”
• For instance:
• A point is a midpoint if and only if it divides a segment into
two congruent segments.
– You can write it as two conditionals that are
converses.
“+”
Converse
=Biconditional
If p then q
+
If q then p
=p if and only if q
p→q
+
q→p
= p ↔ q or q↔p
Conditional
Identifying the Conditional in a
Biconditional
• A ray is an angle bisector if and only if it divides
an angle into two congruent angles.
– Let p and q represent the following:
– p: A ray is an angle bisector
– q: A ray divides an angle into two congruent angles
• p→ q: If a ray is an angle bisector, then it divides an
angle into two congruent angles.
• q → p : If a ray divides an angle into two congruent
angles, then it is an angle bisector.
Examples
• Name the two conditionals that form the
biconditional.
– Two numbers are reciprocals if and only if their
product is 1.
– If two numbers are reciprocals, then their product is 1.
– If the product of two numbers is 1, then the numbers
are reciprocals.
Writing a Biconditional
• What is the converse of the following true
conditional?
– If two angles have equal measure, then the angles
are congruent.
• Converse: If two angles are congruent, then the angles have
equal measure.
– Now Rewrite the statement as a biconditional.
• Two angles have equal measure if and only if the angles are
congruent.
Examples
• Indicate the converse, then write a biconditional
if converse is true.
– If two angles have equal measures, then the angles
are congruent.
– Converse: if two angles are congruent, then the
angles have equal measures; true
– Biconditional: Two angles have equal measures if and
only if the angles are congruent
Real- World Connections
Recap:Summary
• When a conditional and its converse are true,
you can combine them as a true biconditional
using the phrase if and only if.
• The symbolic form of a biconditional is p
can write a good definition as a true
biconditional.
q. You
Ticket Out
Ticket Out
• What is the definition of a
biconditional?
Classwork
• Sect 2.3- Pg 108-109 8,
9, 14, 16, 21,22, 28, 30,
35