1. Geometry
Chapter 2

2. Conditionals A conditional is a statement that can
be written in the If – Then form.
If the team wins the semi-final,
then it will play in the championship.

3. Conditionals The “If” part is called the Hypothesis.
If the team wins the semi-final,
then it will play in the championship.
The “Then” part is called the Conclusion.

4. Hypothesis  Conclusion
Conditionals
Hypothesis  Conclusion
p  q
When
If  Then
All
verbs
Then
Subject adjectives If
are...

5. If Hypothesis  Then Conclusion
Conditionals
Conditional
If Hypothesis  Then Conclusion
p  q
Converse
If  Then
Conclusion
Hypothesis
Hypothesis
Conclusion
q  p

6. Conditionals p  q If the team wins the semi-final,
then it will play in the championship.
q  p
Converse
If the team plays in the championship,
then it won the semi-final.
If it will play in the championship,
then the team wins the semi-final.

7. Conditionals p  q q  p ~ p  ~ q ~ q  ~ p Conditional
If the team wins the semi-final,
then it will play in the championship.
p  q
Converse
If the team plays in the championship,
then it won the semi-final.
q  p
If the team does not win the semi-final,
then it will not play in the
championship.
Inverse
~ p  ~ q
Contrapositive
If the team does not play in the
championship,
then it did not win the semi-final.
~ q  ~ p

8. True False it could be a lion False it could be a lion True
All tigers are cats.
Conditional
If an animal is a tiger,
then it is a cat.
True
p  q
False
Converse
If an animal is a cat,
then it is a tiger.
it could be a lion
q  p
False
Inverse
If an animal is not a tiger,
then it is not a cat.
it could be a lion
~ p  ~ q
Contrapositive
If an animal is not a cat,
then it is not a tiger.
True
~ q  ~ p

9. Postulates Through any two points there exists exactly one line.
If two points exist,
then exactly one line passes through them.

10. Postulates A line contains at least two points. If a line exists,
then it contains at least two points.

11. Postulates
If two lines intersect, then their intersection is exactly one point.

12. Postulates
Through any three noncollinear points there exists exactly one plane.
If three noncollinear points exist,
then exactly one plane contains them.

13. Postulates A plane contains at least three noncollinear points.
If a plane exists,
then it contains at least three noncollinear points.

14. Postulates
If two points lie in a plane, then the line containing them lies in the plane.

15. Postulates
If two planes intersect, then their intersection is a line.

16. Biconditionals True True p  q q  p p  q Conditional
If an angle is acute,
then it has a measure less than 90.
p  q
Converse
If an angle has a measure less than 90,
then it is an acute angle.
True
q  p
If both the conditional and its
Converse are true, then it can
Be written as a biconditional.
Biconditional
p  q
“if and only if”
An angle is acute if and only if
It has a measure less than 90.

17. Practice

18. Notice how the original Conditional has been
Law of Detachment
(1) p  q
Notice how the original Conditional has been
Broken apart
into two pieces.
(Detached)
(2) p
(3) q

19. Law of Detachment p  q p q
If you pass the driving test, then you will get your license. p
Brian passed his driving test. q
Brian got his license.

20. (1) p  q (2) q  r (3) p  r Law of Syllogism Notice how all three
statements are conditionals with three basic ideas.
The repeating part cancels out to give the conclusion.
(2) q  r
(3) p  r

21. Law of Syllogism p  q q  r p  r
If you pass the driving test, then you will get your license.
q  r
If you get your license, then you can drive to school.
p  r
If you pass the driving test, then you can drive to school.

22. If it is not a conditional and not a p statement, Then there is
Working the Laws
If it is not a conditional and not a p statement, Then there is
NO CONCLUSION!
Identify the p and q
If no, is it a p? Then check for Detachment.
If yes, check for syllogism.
Is 2nd statement another conditional ?

23. 3- Joe Nathan has an ID number. Law of Detachment
Examples:
1- If a student is enrolled at Lyons High, then the student has an ID number.
p  q
p: student High
q: student has an ID number
2- Joe Nathan is enrolled at Lyons High. p
3- Joe Nathan has an ID number. q
Law of Detachment

24. NO CONCLUSION Examples: 1- If your car needs more power,
use Powerpack Motor Oil.
p  q
p: car needs more power
q: use Powerpack Motor Oil q
2- Marcus uses Powerpack Motor Oil.
NO CONCLUSION

25. 3- If fossil fuels are burned, then wildlife suffers. Law of Syllogism
Examples:
1- If fossil fuels are burned, then acid rain is produced.
p  q
p: fossil fuels are burned
q: acid rain is produced.
q  r
2- If acid rain falls, wildlife suffers.
3- If fossil fuels are burned,
then wildlife suffers.
p  r
Law of Syllogism

26. INVALID Examples: 1- If a rectangle has four congruent sides,
then it is a square.
p  q
p: a rectangle has four congruent sides
q: it is a square
2- A square has diagonals that are perpendicular.
q  r
3- If a rectangle has four congruent sides,
then its diagonals are perpendicular.
3- A rectangle has diagonals that are perpendicular.
INVALID