1. Section 1.7 and 1.8- Deductive Structure / Statements of Logic
Advanced Geometry
Section 1.7 and 1.8- Deductive Structure / Statements of Logic
Learner Objective: Students will write the converse, inverse, and 
 contrapositive of a conditional statement, determine the truth 
 value of statements and will use the chain rule to make logical 
 conclusions.

2. Learner Objective: Students will write the converse, inverse, and contrapositive of a 
 conditional statement, determine the truth value of statements and will use the 
 chain rule to make logical conclusions. 1 A B C D E

3. Building blocks of Geometry:
 
 
Learner Objective: Students will write the converse, inverse, and contrapositive of a 
 conditional statement, determine the truth value of statements and will use the 
 chain rule to make logical conclusions.
Building blocks of Geometry:
Undefined Terms - Terms which we can't officially 
 define. We describe their properties but don't 
 formally define them.
 examples: point, line, plane
Definitions - State the meaning of a term. There is 
 never a need to prove a definition. (It is true 
 because we say it is true.)
 examples: Def. of a right angle, Def. of a  
 segment bisector...

4. Building blocks of Geometry (cont.):
 
 
Learner Objective: Students will write the converse, inverse, and contrapositive of a 
 conditional statement, determine the truth value of statements and will use the 
 chain rule to make logical conclusions.
Building blocks of Geometry (cont.):
Postulates - An unproved assumption. These are pretty 
 obvious statements which we are unable to prove but 
 which clearly must be true.
 example: Two points determine a line.
Theorems - A mathematical statement which can be 
 proved.
 example: If two angles are both right angles, 
 then they are congruent.

5. If two angles are both right angles,
Learner Objective: Students will write the converse, inverse, and contrapositive of a 
 conditional statement, determine the truth value of statements and will use the 
 chain rule to make logical conclusions.
Definitions, Postulates and Theorems are all conditional 
 statements. They have some (sufficient) condition that 
 leads to a (necessary) conclusion.
If two angles are both right angles,
Having two right angles is sufficient evidence to conclude 
that the angles are congruent.
then they are congruent.
When two angles are both right angles, it is necessary to 
conclude that they are congruent (they have to be).
What happens if we reverse the condition and conclusion?

6. Here is the theorem with the condition and conclusion reversed:
Learner Objective: Students will write the converse, inverse, and contrapositive of a 
 conditional statement, determine the truth value of statements and will use the 
 chain rule to make logical conclusions.
Here is the theorem with the condition and conclusion reversed:
If two angles are congruent
Is knowing that two angles are congruent sufficient 
evidence to conclude that they must be right angles?
then they are both right angles.
If two angles are congruent, is it necessary to conclude 
that they are right angles?
Is this a true statement?
For a statement to be considered true, it must ALWAYS be 
 true. NOT ALWAYS TRUE = FALSE

7. "If p then q" can be symbolized
Learner Objective: Students will write the converse, inverse, and contrapositive of a 
 conditional statement, determine the truth value of statements and will use the 
 chain rule to make logical conclusions.
Conditional statements can be written in "If p then q" form 
where p is the condition or hypothesis of the statement 
(trigger) and q is the conclusion (result).
"If p then q" can be symbolized
(also read "p implies q")

8. Definitions are always reversible.
Learner Objective: Students will write the converse, inverse, and contrapositive of a 
 conditional statement, determine the truth value of statements and will use the 
 chain rule to make logical conclusions.
Reversing the condition and the conclusion results in the 
Converse of the original statement. Thus the convers is 
symbolized
If both the conditional and the converse are true, the 
statement is said to be reversible.
Definitions are always reversible.
Postulates and Theorems are sometimes reversible.

9. Learner Objective: Students will write the converse, inverse, and contrapositive of a 
 conditional statement, determine the truth value of statements and will use the 
 chain rule to make logical conclusions.
The Negation (opposite) of the statement p is "not p" and 
is symbolized
The Inverse of the conditional statement is formed when 
we negate both p and q.
Thus the inverse is symbolized

10. Learner Objective: Students will write the converse, inverse, and contrapositive of a 
 conditional statement, determine the truth value of statements and will use the 
 chain rule to make logical conclusions.
The Contrapositive of the conditional statement is formed 
by both reversing and negating p and q. Thus, the 
contrapositive can be symbolized

11. Conditional T F Converse Inverse Contrapositive
Learner Objective: Students will write the converse, inverse, and contrapositive of a 
 conditional statement, determine the truth value of statements and will use the 
 chain rule to make logical conclusions.
The conditional statement and it's contrapositive have the 
same "truth value". That is, they are either both true or 
both false.
The inverse and the converse have the same truth value.
Conditional T F
Converse
Inverse
Contrapositive

12. If two angles _______________ then ______________________
Learner Objective: Students will write the converse, inverse, and contrapositive of a 
 conditional statement, determine the truth value of statements and will use the 
 chain rule to make logical conclusions.
Example
Conditional:
If two angles are both right angles, 
then they are congruent
Converse:
If two angles _______________
then ______________________
Inverse:
If two angles _______________
then ______________________
Contrapositive:
If two angles _______________
then ______________________

13. Converse: Contrapositive: Example Conditional:
Learner Objective: Students will write the converse, inverse, and contrapositive of a 
 conditional statement, determine the truth value of statements and will use the 
 chain rule to make logical conclusions.
Example
Conditional:
If an angle is an acute angle,
then its measure is greater than 0 and less than 90.
Converse:
If an angle _____________________________,
then __________________________________.
Inverse:
If an angle _____________________________,
then __________________________________.
Contrapositive:
If an angle _____________________________,
then __________________________________.

14. Learner Objective: Students will write the converse, inverse, and contrapositive of a 
 conditional statement, determine the truth value of statements and will use the 
 chain rule to make logical conclusions.
A series of conditional statements can be connected 
together using the Chain Rule. If
and
then

15. Learner Objective: Students will write the converse, inverse, and contrapositive of a 
 conditional statement, determine the truth value of statements and will use the 
 chain rule to make logical conclusions.
Example: , , ,
and

16. Learner Objective: Students will write the converse, inverse, and contrapositive of a 
 conditional statement, determine the truth value of statements and will use the 
 chain rule to make logical conclusions.
At Hilldale High School, there is a rule that any student caught fighting 
must be given a three day suspension. Bill, Bob and Bo are all students 
at Hilldale. Which of the following statements is/are true?
Bill has been given a three day suspension so we know that he 
 must have been caught fighting.
Bob has never been caught fighting, so we know that he has 
 never been given a three day suspension.
Bo has never been given a three day suspension, so we know 
 that he has never been caught fighting.

17. If fighting then suspension.
Learner Objective: Students will write the converse, inverse, and contrapositive of a 
 conditional statement, determine the truth value of statements and will use the 
 chain rule to make logical conclusions.
At Hilldale High School, there is a rule that any student caught fighting 
must be given a three day suspension. Bill, Bob and Bo are all students 
at Hilldale. Which of the following statements is/are true?
Conditional:
If fighting then suspension.
Bill has been given a three day suspension so we know that he 
 must have been caught fighting.
Converse:
If suspension then fighting
False
It is not necessary for someone to Fight in order to receive a 
suspension. Bill could have been suspended for something else. 
The converse is false.

18. If fighting then suspension.
Learner Objective: Students will write the converse, inverse, and contrapositive of a 
 conditional statement, determine the truth value of statements and will use the 
 chain rule to make logical conclusions.
At Hilldale High School, there is a rule that any student caught fighting 
must be given a three day suspension. Bill, Bob and Bo are all students 
at Hilldale. Which of the following statements is/are true?
Conditional:
If fighting then suspension.
Bob has never been caught fighting, so we know that he has 
 never been given a three day suspension.
Inverse:
If NOT fighting then NO suspension.
False
Fighting is a sufficient condition to receive a suspension but not 
a necessary one. Bob could have been suspended for something 
else. The inverse is false.

19. If fighting then suspension.
Learner Objective: Students will write the converse, inverse, and contrapositive of a 
 conditional statement, determine the truth value of statements and will use the 
 chain rule to make logical conclusions.
At Hilldale High School, there is a rule that any student caught fighting 
must be given a three day suspension. Bill, Bob and Bo are all students 
at Hilldale. Which of the following statements is/are true?
Conditional:
If fighting then suspension.
Bo has never been given a three day suspension, so we know 
 that he has never been caught fighting.
If NO suspension then NOT fighting.
Contrapositive:
True
Since suspension is a necessary result of fighting, NOT being 
suspended is sufficient to conclude that there was NO fighting. 
The contrapositive is true.

20. Learner Objective: Students will write the converse, inverse, and contrapositive of a 
 conditional statement, determine the truth value of statements and will use the 
 chain rule to make logical conclusions.

21. Learner Objective: Students will write the converse, inverse, and contrapositive of a 
 conditional statement, determine the truth value of statements and will use the 
 chain rule to make logical conclusions.

22. Learner Objective: Students will write the converse, inverse, and contrapositive of a 
 conditional statement, determine the truth value of statements and will use the 
 chain rule to make logical conclusions.

23. Learner Objective: Students will write the converse, inverse, and contrapositive of a 
 conditional statement, determine the truth value of statements and will use the 
 chain rule to make logical conclusions.

24. Learner Objective: Students will write the converse, inverse, and contrapositive of a 
 conditional statement, determine the truth value of statements and will use the 
 chain rule to make logical conclusions.

25. Learner Objective: Students will write the converse, inverse, and contrapositive of a 
 conditional statement, determine the truth value of statements and will use the 
 chain rule to make logical conclusions.

26. Learner Objective: Students will write the converse, inverse, and contrapositive of a 
 conditional statement, determine the truth value of statements and will use the 
 chain rule to make logical conclusions.

27. Learner Objective: Students will write the converse, inverse, and contrapositive of a 
 conditional statement, determine the truth value of statements and will use the 
 chain rule to make logical conclusions.

28. Learner Objective: Students will write the converse, inverse, and contrapositive of a 
 conditional statement, determine the truth value of statements and will use the 
 chain rule to make logical conclusions.

29. Learner Objective: Students will write the converse, inverse, and contrapositive of a 
 conditional statement, determine the truth value of statements and will use the 
 chain rule to make logical conclusions.

30. Learner Objective: Students will write the converse, inverse, and contrapositive of a 
 conditional statement, determine the truth value of statements and will use the 
 chain rule to make logical conclusions.

31. Learner Objective: Students will write the converse, inverse, and contrapositive of a 
 conditional statement, determine the truth value of statements and will use the 
 chain rule to make logical conclusions.

32. Learner Objective: Students will write the converse, inverse, and contrapositive of a 
 conditional statement, determine the truth value of statements and will use the 
 chain rule to make logical conclusions.

33. Learner Objective: Students will write the converse, inverse, and contrapositive of a 
 conditional statement, determine the truth value of statements and will use the 
 chain rule to make logical conclusions.

34. Learner Objective: Students will write the converse, inverse, and contrapositive of a 
 conditional statement, determine the truth value of statements and will use the 
 chain rule to make logical conclusions.

35. Learner Objective: Students will write the converse, inverse, and contrapositive of a 
 conditional statement, determine the truth value of statements and will use the 
 chain rule to make logical conclusions.

36. Learner Objective: Students will write the converse, inverse, and contrapositive of a 
 conditional statement, determine the truth value of statements and will use the 
 chain rule to make logical conclusions.

37. Learner Objective: Students will write the converse, inverse, and contrapositive of a 
 conditional statement, determine the truth value of statements and will use the 
 chain rule to make logical conclusions.