1. 2.1 Conditional Statements
Geometry
2.1 Conditional Statements

2. November 10, 2018
2.1 Conditional Statements

3. Essential Question When is a conditional statement true or false?
November 10, 2018
2.1 Conditional Statements

4. What you will learn Write conditional statements.
Use definitions written as conditional statements.
Write biconditional statements.
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2.1 Conditional Statements

5. What is a
conditional?
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2.1 Conditional Statements

6. Conditional
A type of logical statement that has two parts, a hypothesis and a conclusion.
A conditional can be written in IF-THEN form.
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2.1 Conditional Statements

7. Shorthand If P, then Q. If HYPOTHESIS , then CONCLUSION.
In the study of logic, P’s and Q’s are universally accepted to represent hypothesis and conclusion.
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2.1 Conditional Statements

8. Example 1 If I study hard, then I will get good grades. I study hard
HYPOTHESIS
CONCLUSION
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2.1 Conditional Statements

9. Can you identify the hypothesis and conclusion?
If today is Monday, then tomorrow is Tuesday.
Hypothesis:
today is Monday
Conclusion:
tomorrow is Tuesday.
Note: IF is NOT part of the hypothesis, and THEN is NOT part of the conclusion.
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2.1 Conditional Statements

10. Your Turn Underline the hypothesis and circle the conclusion.
1. If the weather is warm, then we should go swimming.
2. If you want good service, then take your car to Joe’s Service Center.
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2.1 Conditional Statements

11. Rewriting statements. Use common sense. Don’t over analyze it.
Make sure the sentence is grammatically correct.
The hypothesis always follows “IF.”
No “if?” The first part is usually the hypothesis.
Make your English teacher proud!
Does it sound right?
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2.1 Conditional Statements

12. Example 2a If an animal is a bird, then it has feathers.
Rewrite the following statement in if-then form:
All birds have feathers.
What is the hypothesis?
All birds
What is the conclusion?
have feathers
If-then form?
If an animal is a bird, then it has feathers.
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2.1 Conditional Statements

13. Example 2b If you are in Houston, then you are in Texas.
Rewrite the following statement in if-then form:
You are in Texas if you are in Houston.
What is the hypothesis?
You are in Houston
What is the conclusion?
You are in Texas
If-then form?
If you are in Houston, then you are in Texas.
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2.1 Conditional Statements

14. Example 2c If a number is even, then it is divisible by 2.
Rewrite the following statement in if-then form:
An even number is divisible by 2.
What is the hypothesis?
An even number
What is the conclusion?
Divisible by 2.
If-then form?
If a number is even, then it is divisible by 2.
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2.1 Conditional Statements

15. Your TUrn Rewrite the conditional statement in if-then form.
3. Today is Monday if yesterday was Sunday.
If yesterday was Sunday, then today is Monday.
4. An object that measures 12 inches is one foot long.
If an object measures 12 inches, then it is one foot long.
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2.1 Conditional Statements

16. Negation The negative of the original statement. Examples: I am happy.
I am not happy.
mC = 30°.
mC  30°.
A, B and C are on the same line.
A, B and C are not on the same line.
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2.1 Conditional Statements

17. Negation
Think NOT
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2.1 Conditional Statements

18. Example 3 Write the negation of each statement. a. The ball is red.
The ball is not red.
b. The cat is not black.
The cat is black.
c. The car is white.
The car is not white.
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2.1 Conditional Statements

19. Related
Conditionals
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2.1 Conditional Statements

20. Related conditional statements
Looking at the conditional statement: If p, then q.
There are three similar statements we can make.
Converse
Inverse
Contrapositive
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2.1 Conditional Statements

21. Converse
If Q, then P.
The converse of a statement is formed by switching the hypothesis and the conclusion.
Conditional:
If you play drums, then you are in the band.
Converse: If
you are in the band,
then
you play drums.
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2.1 Conditional Statements

22. Example 4 Write the converse of the statement below.
If you like tennis, then you play on the tennis team.
Answer:
If you play on the tennis team, then you like tennis.
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2.1 Conditional Statements

23. Inverse
If not P, then not Q.
The inverse is formed by taking the negation of the hypothesis and of the conclusion.
Conditional:
If x = 3, then 2x = 6.
Inverse:
If x  3, then 2x  6.
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2.1 Conditional Statements

24. Example 5 Write the inverse of the statement below.
If today is Monday, then tomorrow is Tuesday.
Answer:
If today is not Monday, then tomorrow is not Tuesday.
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2.1 Conditional Statements

25. Contrapositive
If not Q, then not P.
The contrapositive is formed by switching and negating the hypothesis and the conclusion.
(Take the inverse of the converse, or, the converse of the inverse.)
Conditional:
If I am in 10th grade, then I am a sophomore.
Contrapositive:
If I am not a sophomore, then I am not in 10th grade.
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2.1 Conditional Statements

26. Example 6 Write the contrapositive of the statement below.
If x is odd, then x + 1 is even.
Negate
Negate
x + 1 is not even
x is not odd
If x+1 is not even,
then x is not odd.
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2.1 Conditional Statements

27. Logical Statements If I live in Mesa, then I live in Arizona.
Converse:
Inverse:
Contrapostive:
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2.1 Conditional Statements

28. Warm up Day 2 Find the distance between the given points.
1. (𝟑, 𝟖) 𝒂𝒏𝒅 (−𝟐, 𝟕) (𝟓, 𝟒) 𝒂𝒏𝒅 (−𝟏, −𝟑)
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2.1 Conditional Statements

29. Your TUrn. Write the converse, inverse, and contrapositive.
If mA = 20, then A is acute.
Converse: (switch hypothesis and conclusion)
If A is acute, then mA = 20.
Inverse: (negate hypothesis and conclusion)
If mA  20, then A is not acute.
Contrapositive: (switch and negate both)
If A is not acute, then mA  20.
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2.1 Conditional Statements

30. Review: Logical Statements
Conditional: If P, then Q.
Converse: If Q, then P.
Inverse: If not P, then not Q.
Contrapositive: If not Q, then not P.
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2.1 Conditional Statements

31. Definition: Perpendicular Lines
Two lines that intersect to form a right angle. m n
Notation:
m  n
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2.1 Conditional Statements

32. Using definitions The conditional statement would be:
You can write a definition as a conditional statement in if- then form. Let’s look at an example:
The conditional statement would be:
The converse statement also ends up being true:
Perpendicular Lines: two lines that intersect to form a right angle.
If two lines are perpendicular, then they intersect to form a right angle.
If two lines intersect to form a right angle, then they are perpendicular lines.
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2.1 Conditional Statements

33. 2.1 Conditional Statements
Day 2
2.1 Conditional Statements

34. Truth Values A conditional is either True or False.
To show that it is true, you must have an argument (a proof) that it is true in all cases.
To show that it is false, you need to provide at least one counterexample.
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2.1 Conditional Statements

35. Example 7 True or false? If false provide a counter example.
If x2= 9, then x = 3.
FALSE!
Counterexample: x could be –3.
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2.1 Conditional Statements

36. True! Example 8 If x = 10, then x + 4 = 14. Proof: x = 10
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2.1 Conditional Statements

37. Equivalent Statements
When two statements are both true or both false, they are called equivalent statements.
A conditional statement is always equivalent to its contrapositive.
The inverse and converse are also equivalent.
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2.1 Conditional Statements

38. Equivalent Statements
Original:
If mA = 20, then A is acute.
Converse: (switch hypothesis and conclusion)
If A is acute, then mA = 20.
Inverse: (negate hypothesis and conclusion)
If mA  20, then A is not acute.
Contrapositive: (switch and negate both)
If A is not acute, then mA  20.
TRUE
False
False
TRUE
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2.1 Conditional Statements

39. Example 9 Statement: If x = 5, then x2 = 25. TRUE
Contrapositive: If x2  25, then x  5.
TRUE
Converse: If x2 = 25, then x = 5.
FALSE – could be –5.
Inverse: If x  5, then x2  25.
FALSE
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2.1 Conditional Statements

40. Justifying Statements
In math, deciding if a statement is true or false demands that you can justify your answers. “Just because”, or, “It looks like it” are not sufficient.
Justification must come in the form of Postulates, Definitions, or Theorems.
November 10, 2018
Geometry 2.1 Conditional Statements

41. Example 10 Statement A D, X, and B are collinear. Truth Value D X B
TRUE
Reason
Definition of collinear points. C
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Geometry 2.1 Conditional Statements

42. Example 11 Def  lines Statement A AC  DB Truth Value D X B TRUE
Reason
Definition of Perpendicular lines
Def  lines C
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Geometry 2.1 Conditional Statements

43. Example 12 CXB is adjacent to BXA Def. of adjacent angles
Statement
CXB is adjacent
to BXA A
Truth Value D X B
TRUE
Reason
Def. of adjacent angles C
Def. of adj. s
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Geometry 2.1 Conditional Statements

44. Example 13 Statement DXA and CXB are adjacent angles. A Truth Value
FALSE
Reason
There is not a common side. (Or, they are vertical angles.) C
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Geometry 2.1 Conditional Statements

45. VERY IMPORTANT!
In doing proofs, you must be able to justify every statement with a valid reason. To be able to do this you must know every definition, postulate and theorem. Being able to look them up is no substitute for memorization.
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Geometry 2.1 Conditional Statements

46. Your Turn D A H E B C G F November 10, 2018
3rd hour
Your Turn D A H E B C F G
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Geometry 2.1 Conditional Statements

47. Your Turn D A False (they are not collinear) H E B CD A
False (they are not collinear) H E B
True (sides are opposite rays) C F G
True (post. 8)
False (no rt.  mark)
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Geometry 2.1 Conditional Statements

48. Your Turn True (def.  lines) D A False (they are supplementary) H E B
True (def.  lines) D A
False (they are supplementary) H E B C
True (half of 180 is 90 -- a right ) F G
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Geometry 2.1 Conditional Statements

49. Biconditionals
When a conditional statement and its converse are both TRUE, they can be written as a single biconditional statement. Let’s look at an example:
Conditional
If 2 s are complementary, then their sum is 90°.
True
Converse
If the sum of 2 s is 90°, then they are complementary.
True
Biconditional
2 s are complementary if and only if their sum is 90°.
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2.1 Conditional Statements

50. Biconditionals (Continued)
Written with p’s and q’s a biconditional looks like this:
p if and only if q. or
p iff q.
Iff means “if and only if”.
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2.1 Conditional Statements

51. Putting it all together
Statements
In words
In symbols
Conditional
If p, then q
𝑝→𝑞
Converse
If q, then p
𝑞→𝑝
Inverse
If not p, then not q
~𝑝→~𝑞
Contrapostive
If not q, then not p
~𝑞→~𝑝
Biconditional
p if and only if q
𝑝↔𝑞
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2.1 Conditional Statements

52. Example 14 Let P be the statement: “x = 3”
Let Q be the statement: “2x = 6”
Write:
P  Q
Q  P
P  Q
If x = 3, then 2x = 6.
If 2x = 6, then x = 3.
x = 3 if and only if 2x = 6.
or 2x = 6 iff x = 3.
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2.3 Deductive Reasoning

53. Definitions ALL definitions are biconditionals. Example:
Definition of Congruent Angles
Two angles are congruent iff they have the same measure.
Conditional: If two angles are congruent, then they have the same measure.
Converse: If two angles have the same measure, then they are congruent.
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Geometry 2.1 Conditional Statements

54. Truth Values of Biconditionals
A biconditional is TRUE if both the conditional and the converse are true.
A biconditional is FALSE if either the conditional or the converse is false, or both are false.
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Geometry 2.1 Conditional Statements

55. Example 15 False! Biconditional x = 5 iff x2 = 25. Conditional
If x = 5, then x2 = 25.
Converse
If x2 = 25, then x = 5.
False!
True or False?
True or False?
true
True or False?
False!
November 10, 2018
Geometry 2.1 Conditional Statements

56. Your Turn
Write the following biconditional statement as a conditional statement and its converse.
An angle is obtuse iff it measures between 90 and 180.
Answer
Conditional: If an angle is obtuse, then it measures between 90 and 180.
Converse: If an angle measures between 90 and 180, then it is obtuse.
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Geometry 2.1 Conditional Statements

57. Why is this important? Geometry is stated in rules of logic.
We use logic to prove things.
It teaches us to think clearly and without error.
It impresses girl friends (or boy friends).
You can talk like…
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2.1 Conditional Statements

58. November 10, 2018
2.1 Conditional Statements

59. Assignment 2.1 #2-20 even, 26-38 even, 50, 55, 58 November 10, 2018
2.1 Conditional Statements