Download UNIT 2 NOTES Geometry A Lesson 7 – Inductive Reasoning Can

UNIT 2 NOTES
Lesson 7 – Inductive Reasoning
Geometry A
1. I CAN understand what inductive reasoning is and its importance in geometry
3. I CAN show that a conditional statement is false by finding a counterexample
Can you find the next item in each pattern?
▪ Monday, Wednesday, Friday, _______________
▪ 3, 6, 9, 12, 15, ____
▪ ←, ↑, →, ____
▪ 0.4, 0.04, 0.004, 0.0004, ________
In the examples above, you used a process called inductive reasoning to continue the pattern.
Inductive reasoning is ________________________________________________________
___________________________________________________________________________
A conjecture is ______________________________________________________________
Examples: complete each conjecture
“The quotient of two positive numbers is ____________.”
“The number of lines formed by four points, no three of which are collinear, is _____.”
“The product of two odd numbers is ________.”
Average Whale Lengths in feet
Length of female (ft) 49 51 50 48 51 47
Length of male (ft)
47 45 44 46 48 48
UNIT 2 NOTES
To show that a conjecture is TRUE, you must prove it (more on that later).
Geometry A
To show that a conjecture is FALSE, you have to find a counterexample, which is _________
___________________________________________________________________________
Inductive reasoning process:
1. Look for a _______________.
2. Make a _______________.
3. __________ the conjecture or find a ____________________.
Ex: Show that each conjecture is false by finding a counterexample
“For every integer n, the value of n3 is
positive.”
“Two complementary angles are not
congruent.”
-----------------------------------------------------------------------------------------------------------------------------PRACTICE – complete the following the best you can
1. Find the next item in each pattern:
March, May, July, _______________
1 2 3
, , ,
3 4 5
2. Complete the conjecture: “The product of two even numbers is ________.”
3. Show that each conjecture is false by finding a counterexample:
“Three points on a plane always form a
triangle.”
“For any real number x, if x2 ≥ 1, then x ≥ 1.
UNIT 2 NOTES
Lesson 8 – Conditional Statements
Geometry A
2. I CAN recognize and write conditional statements and determine their truth value
3. I CAN show that a conditional statement is false by finding a counterexample
A conditional statement is a statement that can written in the form “If __ , then_ .”
Symbols:
The hypothesis is
.
The conclusion is
.
Example 1:
a.
If it is raining, then the sidewalks are wet.
Hypothesis:
Conclusion:
b.
A number is a rational number, if it is an integer.
Hypothesis:
Conclusion:
Even though some statements don’t have the words “If” and “then”, they are still conditional
statements, if one statement depends on the other.
Write the statement in “If-then” form.
A right triangle is a triangle with one right angle.
If
, then
.
We have early-release days on Wednesdays.
If
, then
.
Every conditional statement that is made is either true or false.
The only way a statement can be considered false is if the hypothesis is true, and the
conclusion is false.
p
q
pq
*Remember, to show that something is false,
you only need to provide one
counterexample where the hypothesis is
true, and conclusion is false.
UNIT 2 NOTES
Example 3: Determine whether true or false.
Geometry A
1. If the month is August, then the next month is September.
2. If two angles are acute, then they are congruent.
3. If 4 is prime, then 5  4  8 .
The negation of statement p, written as
.
If the statement is “It is raining”, the negation of that statement would be
.
Negations are used to write related conditionals.
Related Conditionals
TERM
STATEMENT
SYMBOLS
CONDITIONAL
CONVERSE
INVERSE
CONTRAPOSITIVE
Example 4:
Write the converse, inverse, and contrapositive. Then find its truth value.
TERM
STATEMENT
CONDITIONAL
If an animal is a cat, then it has four paws.
Truth
Value
CONVERSE
INVERSE
CONTRAPOSITIVE
Notice that the some of the related conditionals have the same truth value. These are called
statements.
and
are always logically equivalent.
and
are always logically equivalent.
UNIT 2 NOTES
Lesson 9 – Deductive Reasoning
Geometry A
4. I CAN understand what deductive reasoning is and its importance in geometry
5. I CAN determine the validity of conclusions using the Laws of Detachment and Syllogism
Deductive reasoning is
In deductive reasoning, if given facts are true and you apply correct logic, the conclusion must
be true.
Two basic laws that are used to apply logic:
Law of Detachment:
Example 1: According to the Law of Detachment, is the conjecture valid?
Given:
In the World Series, if a team wins four games, then the team wins
the series.
The Red Sox won four games in the World Series in 2004.
Conjecture: The Red Sox won the World Series.
Given:
If you are tardy 3 times, you must go to detention.
George is in detention.
Conjecture: George was tardy 3 times.
Given:
If a student passes his classes, the student is eligible to play sports.
Ramon passed his classes.
Conjecture: Ramon is eligible to play sports.
Given:
If it is Halloween, then Sheila wears a costume. Sheila is wearing a
costume.
Conjecture: Today is Halloween.
Given:
If you want the best hamburger there is, then you go to Red Robin.
Brandon went to Red Robin.
Conjecture: Brandon had the best hamburger there is.
.
UNIT 2 NOTES
Geometry A
Law of Syllogism:
Example 2: Use the Law of Syllogism to determine if the conjecture is valid
Given
If a figure is a kite, then it is a quadrilateral. If a figure is a
quadrilateral, then it is a polygon. Figure WXYZ is a kite.
Conjecture: Figure WXYZ is a polygon.
Given:
If a number is divisible by 2, then it is even. If a number is even, then
it is an integer. A number is an integer.
Conjecture: It is divisible by 2.
Example 3: Use the Law of Syllogism to draw a conclusion
Given:
If you attend GLHS, then you are a Blue Devil. If you are a Blue Devil, then you
are awesome.
Conclusion: If you attend GLHS, _________________________________________________
Given:
If it is autumn, the leaves change color. If it is October, then it is autumn.
Conclusion: _________________________________________________________________
UNIT 2 NOTES
Lesson 10 – Deductive Reasoning
6. I CAN write a definition as a biconditional
7. I CAN determine if a biconditional statement is true or false
A biconditional statement is…
Write the biconditional as its conditional and converse.
A.
An angle is a right angle if and only if it measures 90°.
B.
A solution is neutral iff it’s pH is 7.
Write the converse and biconditional from the conditional.
A. If 5x  8  37 , then x  9 .
B.
If you live in Hell, then you live in Michigan.
As you can see, sometimes a conditional and its converse are not both true.
For a biconditional to be true, its conditional AND converse must be true.
If either statement is false, then the biconditional is false.
In other words, a biconditional statement must be true both ways that is read.
Geometry A
UNIT 2 NOTES
Geometry A
Write the conditional and converse from the biconditional. Then, determine if the
biconditional is true or false.
A. A rectangle has side lengths of 10 cm and 30 cm if and only if its area is 300 cm 2.
B. y  5  y 2  25
C. An angle has a measure of 180° if and only if it is a straight angle.
In geometry, biconditionals are used to write definitions. A definition is valid if it can written as a
true biconditional.
Write the definition of vertical angles as a biconditional.
Vertical angles are two non-adjacent angles formed by intersecting lines.
Write the definition of binomial as a biconditional.
A binomial is an algebraic expression with exactly two terms.
UNIT 2 NOTES
Lesson 11 – Introduction to Proofs
Geometry A
9. I CAN justify a statement using a property, definition, postulate, or theorem
10. I CAN write a deductive proof involving lines, segments, and angles
In this lesson, we are going to demonstrate the process of doing a geometric proof.
▪ Each STATEMENT in a proof must follow logically from what has come before and must have
a reason to support it.
▪ The REASON may be a piece of given information, a definition, a postulate, a property, or a
previously proved theorem.
▪ The idea is straightforward…start with one or more given facts, apply a logical chain of
reasoning, and end with a conclusion
EXAMPLE
Given:
Prove:
B is the midpoint of AC
AB  EF
BC  EF
Statement
Reason
1.
2.
3.
4.
5.
6.
EXAMPLE
Given:
Prove:
A and B are complementary
A  C
B and C are complementary
Statement
1.
2.
3.
4.
5.
6.
Reason
UNIT 2 NOTES
Geometry A
EXAMPLE
Given:
mA  60
mB  2mA
A and B are supplementary
Prove:
Statement
1.
2.
3.
4.
5.
6.
mA  60 ; mB  2mA
mB  2(60 )
mB  120
mA  mB  60  120
mA  mB  180
A and B are supplementary.
EXAMPLE
Given:
Prove:
X is the midpoint of AY
Y is the midpoint of XB
AX  YB
Statement
1.
2.
3.
4.
5.
6.
Reason
Reason
UNIT 2 NOTES
Lesson 12 – Proving Geometric Theorems
Geometry A
10. I CAN write a deductive proof involving lines, segments, and angles
11. I CAN prove the following theorems: Right Angle Congruence, Vertical Angles, and Linear Pair
Right Angle Congruence Theorem
1 and 2 are right angles
Given:
1  2
Prove:
Statement
Reason
1.
2.
3.
4.
5.
Linear Pair Theorem
1 and 2 are a linear pair
Given:
1 and 2 are supplementary
Prove:
Statement
Reason
1.
2.
3.
4.
5.
6.
Vertical Angles Theorem
1 and 3 are vertical angles
Given:
Prove:
1  3
Statement
1.
2.
3.
4.
5.
6.
7.
8.
9.
Reason
UNIT 2 NOTES
Lesson 13 – Two Lines Cut by a Transversal
Geometry A
12. I CAN recognize angle pairs when two lines are cut by a transversal
A transversal is a line that intersects two coplanar lines at two different points. The transversal
t and the other two lines r and s always form eight angles.
Term
Definition
Corresponding
Angles
Alternate
Interior
Angles
Alternate
Exterior
Angles
Same-side
Interior
Angles
Ex 2:
Classifying Pairs of Angles
a. corresponding angles
b. alt. exterior angles
c. alt. interior angles
d. same-side interior angles
Ex. 3:
Identifying Angle Pairs and Transversals
a. 1 and 3
b. 2 and 6
c. 4 and 6
Example
UNIT 2 NOTES
Lesson 14 – Proving Parallel Lines Theorems
Geometry A
12. I CAN recognize angle pairs when two lines are cut by a transversal
13. I CAN prove and apply the following theorems: Corresponding Angles, Alternate Interior Angles, Alternate
Exterior Angles, and Same-Side Interior Angles
In the previous lesson, we have already made conjectures about what happens to the angle
pairs when we have two parallel lines and a transversal. We are going to prove the theorems
below.
In order to prove these theorems, one of the relationships must be accepted without proof as a
postulate. It can then be used to prove the other relationships. We are going to accept
corresponding angles as a postulate.
Proof of the ALTERNATE INTERIOR ANGLES THEOREM
Given: 2 and 3 are alternate interior angles
1
2
3
Prove: 2  3
Statement
Reason
1. 2 and 3 are alternate interior angles
2. 1  3
3. 1 and 2 are vertical angles
4. 1  2
5. 2  3
Proof of the SAME-SIDE INTERIOR ANGLES THEOREM
Given: 8 and 9 are same-side interior angles
7
8
9
Prove: 8 and 9 are supplementary
Statement
1. 8 and 9 are same-side interior angles
2.
3. m7  m9
4.
5. 7 and 8 are supplementary
6.
7. m8  m9  180
8.
Reason
Corresponding Angles Postulate
Definition of a Linear Pair
Definition of Supplementary Angles
Definition of Supplementary Angles
UNIT 2 NOTES
Geometry A
Practice:
Use the parallel line theorems to find the missing angle. Be sure to state which
theorem/postulate you used.
1. Find mABD .
2. Find mTUS .
3. Find mABC .
4. Find the value of x.
UNIT 2 NOTES
Lesson 15 – Proving Lines Parallel
Geometry A
12. I CAN recognize angle pairs when two lines are cut by a transversal
14. I CAN prove and apply the following theorems: Converse of Corresponding Angles, Converse of Alternate
Interior Angles, Converse of Alternate Exterior Angles, and Converse of Same-Side Interior Angles
We have already established the angle relationships that occur when we have parallel lines.
What about the converse of these theorems and postulate? Are they true?
These have been proven – you are going to write two of the proofs for homework.
Proof of the Converse of the Alternate Interior Angles Theorem
1  2
Given:
Prove:
3
n m
Statement
Reason
1.
2.
3.
4.
Whereas the previous lesson gave us parallel lines so that we could use the angle pair
relationships to solve equations, this lessons gives us angle pair relationships so that we can
determine if the lines are parallel or not.
▪ If one of the boxes on your “Parallel Lines Theorems – Converses” sheet is satisfied, then the
lines are PARALLEL.
▪ If one of the boxes on your “Parallel Lines Theorems – Converses” sheet is contradicted, then
the lines are NOT PARALLEL.
▪ If the given information doesn’t fit at all into one of the boxes on your “Parallel Lines Theorems
– Converses” sheet, then there is NOT ENOUGH INFORMATION to tell if the lines are parallel
or not.
Use the three statements above to do the example problems that follow on the back…
UNIT 2 NOTES
Geometry A
Determine if the following relationships prove that the lines are parallel. If so, you must have a
theorem/postulate that supports it. If not, determine if there is “not enough info” or “not parallel”.
1. m2  150 , m7  150
2. 3  6
3. m1  m5
4. 6  7
5. 3  8
6. m6  100 , m8  80
7. m3  45 , m5  135
8. 1  8
9. m2  50 , m6  60