Biconditional Statements

1
Section 2-2

• Biconditional Statements
• And Definitions

2
Recall

• This is a Conditional Statement
• If two angles have the same measure, then the
  angles are congruent.
• This is its Converse
• If two angles are congruent, then the angles have
  the same measure.
• Is the Conditional true?
• Is the Converse true?

Yes
Yes
3
Biconditional or Equivalence Statements

• If both the conditional and its converse are
  true, you can combine them as a true
  biconditional or equivalence statement
• Two angles have the same measure if and only if
  the angles are congruent.
• Note Sometimes iff is used for if and only if

4
In Symbols

• We combine
• As

and
5
Separating Biconditionals

• You can separate a biconditional into 2
  conditionals that are converses of each other.
• A shape is a pentagon if and only if it has 5
  sides.
• If a shape is a pentagon, then it has 5 sides.
• If a shape has 5 sides, then it is a pentagon.

6
Recognizing Good Definitions

• A good definition
• Uses clearly understood terms
• Is precise (avoids words like large, sort of, and
  some)
• Can be written as a true biconditional

7
Show that this is a good definition

• A right angle is an angle whose measure is 90
  degrees.
• Conditional
• If an angle is a right angle, then it measures 90
  degrees.
• Converse
• If an angle measures 90 degrees, then it is a
  right angle.
• Both the conditional and converse are true so we
  can write the biconditional
• An angle is a right angle if and only if its
  measure is 90 degrees.

8
Why are these poor definitions?

• (Hint Find a counterexample)
• An elephant is a large animal.
• A square is a shape with four sides.
• An airplane is a vehicle that flies.