1. Biconditionals Goal: To learn about and use biconditional statements
And definitions may 15/16

2. Contrapositive: does both
Let’s Re-Cap!
CONDITIONAL statements can be written in the form “if p, then q.” The CONVERSE states, “if q, then p” The INVERSE states “if not p, then not q.” The CONTRAPOSITIVE states “if not q, then not p.
Converse: flips it
Inverse: negates it
Contrapositive: does both

3. Biconditional statements
“You can go to the picture show IF AND ONLY IF you clean up this room, Sonny!”
What this means...
If clean room  then go to show
AND
If go to the show  then clean room
**Tips**
The words, “only if” don’t count as a biconditional… it’s the same as “if.”
The abbreviation for a biconditional is p q.

4.  EXAMPLES: Going backward Going forward 1.
If a polygon is a Δ, then it has exactly 3 sides.
If a polygon has exactly 3 sides, then it is a Δ.
Going backward
Going forward 2. 
If an < is right , then it measures 90°.
If an < measures 90°, then it is right.
Tip: Don’t move the word “if,” only switch the hypothesis and conclusion.

5. EXAMPLE: Determine if the biconditional is true.
An angle is straight iff its measure is 180 degrees.
This means:
An angle is straight if its measure is 180 degrees and
An angle’s measure is 180 degrees if it is a straight angle.
Are each of these statements true?
YES! So the biconditional is TRUE!
“if and only if”

6. A number is divisible by 5 iff it ends in 0.
EXAMPLE: Determine if the biconditional is true.
Re-write the biconditional as two conditional statements.
Determine whether the biconditional is true.
If not, provide a counterexample.
A number is divisible by 5 iff it ends in 0.
If a # is ÷ by 5, then it ends in 0.
If a # ends in 0, then it is ÷ by 5.
FALSE (not always true)
TRUE
If it is FALSE in one direction, then the entire biconditional is FALSE!
False, Counterexample: 15 is ÷ by 5, but does not end in 0.

7. YOU TRY!: Determine if the biconditional is true.
Re-write the biconditional as two conditional statements.
Determine whether the biconditional is true.
If not, provide a counterexample.
A line bisects a segment iff it intersects the segment at its midpoint.
If a line bisects a segment, then it intersects at its midpoint.
If a line intersects a segment at its midpoint, then it bisects it.
TRUE
TRUE
The biconditional is TRUE!

8. YOU TRY! Determine if the biconditional is true.
Re-write the biconditional as two conditional statements.
Determine whether the biconditional is true.
If not, provide a counterexample.
x =10 iff |x| = 10 .
If x = 10, then |x| = 10.
If |x| = 10, then x = 10.
TRUE
FALSE...(not the whole truth)
x = 10 or -10.
FALSE! Counterexample: x = -10.

9. ✗ ✗ ✗ ✔ WRITING DEFINITIONS: A fish is an animal that swims.
Giraffes are animals with very long necks.
An American penny is worth one cent.
A fish is an animal that swims.
Rectangles have four corners. ✗ ✗ ✗ ✔

10. Determine if the following are good definitions. If not, explain.
A square is a polygon that has 4 equal sides.
Two angles that form a linear pair are both adjacent and supplementary.
Opposite rays are two rays that share the same endpoint.
No! A rhombus has 4 = sides, but is not a square.
A square has 4 equal sides AND 4 right angles.
No! These rays share the same endpoint but are not opposite.
Yes!
Opposite rays share an endpoint AND form a straight line.

11. ✗ ... x can be <0, and |x| will still be >0.
More Practice
✗ ... x can be <0, and |x| will still be >0.
✗ ... x can be -3, and x2 will still = 9.
✗ does not = 5.
✔ ... x can only be 19, if 2x – 3 = 35, and vise versa.