Download Geometry 1 Unit 2 Reasoning and Proof Student Notes

Name ___________
Geometry 1
Unit 2:
Reasoning and Proof
August 25
2.1 Conditional
Statements
26
27
2.2 Definitions and
Biconditional
Statements
2.1 Conditional
Statements
Page 75 #10 - 16
even, Page 77 #50,
Page 78 #62, 64, 72, Pages 76 - 77 #18 74
32 even, 40-44 all,
28
2.3 Deductive
Reasoning
29
2.3 Deductive
Reasoning
Pages 92 - 93 #22 - 30
Pages 82 - 83 14 - 28
even, #34 - 40 even,
even, 40, 41, 42
Pages 91 #8 - 14 all, 50
September
Monday
Tuesday
1
2
NO SCHOOL
Review 2.1-2.3
Labor Day
ASVAB
testing in Sept.
Wednesday
Page 94 # 56 - 63,
Page 95 #1 - 5
8
Thursday
4 Progress Rpts Due
3
2.4 Reasoning and
Properties from
Algebra
Friday
2.5 Proving
Statements about
Segments
2.5 Proving
Statements about
Segments
Page 99 - 100 #10 16all, page 103 # 40, Page 105-107 #6, 7,
42, 45
16, 19, 31-34
Page 105 #8-11all
9
10
11
2.6 Proving
Statements about
Angles
2.6 Proving
Statements about
Angles
Page 113 #10- 18
even,
Page 114 #20, 22, 27,
28
Page 121 #1 - 19
Review Unit 2
Review Unit 2
Study
5
12
Geometry 1
Chapter 2 Reasoning
and Proof Exam
1
Write the directions for making a Peanut Butter and Jelly Sandwich.
2
Geometry 1
2.1: Conditional statements
Conditional Statement
Unit 2: Reasoning and Proof
If-then form
Hypothesis
Conclusion
Examples of
Conditional
Statements
Example 1
Circle the hypothesis and underline the conclusion in each conditional
statement
Example 2
A line contains at least two points.
When two planes intersect their intersection is a line.
Two angles that add to 90° are complementary.
3
Label the following statement as true or false. If the statement is false, then give a
counterexample.
If it is Monday, then it is a school day.
Write a true statement.
Write a false statement and provide a counterexample to show that it is false.
4
Counterexample
Example 3
Determine if the following statements are true or false.
If false, give a counterexample.
If x + 1 = 0, then x = -1
If a polygon has six sides, then it is a decagon.
If the angles are a linear pair, then the sum of the measure of the angles is
90º.
Negation
Examples of
Negations
Example 4
Write the negation of each statement. Determine whether your new
statement is true or false.
Yuma is the largest city in Arizona.
All triangles have three sides.
Dairy cows are not purple.
Some CGUHS students have brown hair.
5
Write the following as an if- then statement, then write the converse, inverse, and
contrapositive of the conditional statement.
“A line contains at least two points.”
Statement:
Converse:
Inverse:
Contrapositive:
Is the statement true?
Is the converse true?
Is the inverse true?
Is the contrapositive true?
6
Converse
Inverse
Contrapositive
Examples


Write in if…then form.
Write the converse, inverse and contrapositive of each statement.
I will wash the dishes, if you dry them.
If…then form
Converse
Inverse
Contrapositive
A square with side length 2 cm has an area of 4 cm2.
If…then form
Converse
Inverse
Contrapositive
7
Sketch an example to represent each of the 5 postulates learned in this lesson:
1.
2.
3.
4.
5.
8
Point-line Postulate
Point-line converse
Intersecting lines
postulate
Point-plane postulate
Point-plane converse
Line-plane postulate
Intersecting planes
postulate
9
Determine whether each statement and its converse are true or false. If the statement
and the converse are true, write as a biconditional statement, if false, write a
counterexample.
1.
If an angle measures 90 degrees, then it is a right angle
2.
If it is Saturday, then there is no school
3.
If you do your homework, then you will pass the class.
4.
If a figures contains at least two points, then the figure is a line
10
Geometry 1
2.2- Biconditional Statements
Biconditional Statement
Example 1
Unit 2: Reasoning and Proof
Determine whether the biconditional is true
Perpendicular Lines
A line perpendicular to a
plane

Example 2
Example 3
Example 4
Example 5
11
Use symbols to represent (a) a conditional statement, (b) the converse, (c) the inverse,
and (d) the contrapositive.
a)
b)
c)
d)
Write a statement Similar to example 2 on pg. 13, then make up 3 problems for that
statement
Statement:
Example: Write p ^ q for the given statement
1.
2.
3.
12
Geometry 1
2.3: Deductive Reasoning
Symbolic logic
Unit 2: Reasoning and Proof
~
Symbols
→
Example 1
Let p be “the measure of two angles is 180º
Let q be “two angles are supplementary
p→q
q→p
Example 2
P: Jen Cleaned her room
q: Jen is going to the mall
What does p→q mean?
What does q→p mean?
What does ~q mean?
What does p ^ q mean?
Example 3
t: Jeff has a math test today
s: Jeff Studied
t s
s→t
~s→t
What does ~q mean?
13
If Ana completes all of her homework, then she will go to the movies
Ana completed all of her homework
What will Ana do now?
If Joe wins the football game, he will get a new movie
John did not win the football game
Will John get a new movie?
Law of Syllogism
If Derrick cleans his room, he will go to the mall
If Derrick goes to the mall, he will get new shoes
Derrick cleaned his room, does he get new shoes?
Write your own system of expressions using the law of syllogism
14
Deductive Reasoning
Law of Detachment
Example 4
Determine if the argument is valid
Example 5
Determine if the argument is valid
Law of Syllogism
Example 6
Example 7
15
Write an example to illustrate each of the algebraic properties of equality.
Addition Property
Subtraction Property
Multiplication Property
Division Property
Reflexive Property
Symmetric Property
Transitive Property
Substitution Property
Distributive Property
16
Geometry 1
2.4: Reasoning with properties from Algebra
Objectives:
Unit 2: Reasoning and Proof
1.
2.
3.
Algebraic properties of equality
Addition property
Subtraction property
Multiplication property
Division property
Reflexive property
Symmetric Property
Transitive property
Substitution Property
Distributive Property
Example 1
Statement
Reason
17
Create an algebraic equation. Then solve the equation step by step and write the
reason for each step.
Equation:
Step
1.
Reason
1.
2.
2.
3.
3.
4.
4.
5.
5.
6.
6.
7.
7.
18
Example 2:
Solve 2(x – 3) = 6x + 6
Statement
Valid or invalid algebraic
equations
Reason
Determine if the equations are valid or invalid.
(x + 2) (x + 2) = x2 + 4
X3x3 = x6
– (x + y) = x – y
Geometric Properties of
Equality
Reflexive property
Symmetric Property
Transitive Property
Example 3
In the diagram, AB = CD. Show that AC = BD
A
Statement
B
Reason
C
D
AB = CD
AB + BC = BC + CD
AC = AB + BC
BD = BC + CD
AC = BD
19
Write an example of each of the three properties of segment congruence, and draw an
example that illustrates each of these geometric properties of equality
1.
2.
3.
20
Geometry 1
2.5: Proving Statements about line segments
Unit 2: Reasoning and Proof
2-column proof
Theorem
Properties of Segment
congruence
Reflexive
Symmetric
Transitive
Example 1
Triangle segment proof
In Triangle JKL,
Given: LK = 5, JK = 5, JK = JL,
Prove: LK = JL
Statement
Reason
1.
1.
2.
2.
3.
3.
4.
4.
5.
5.
6.
6.
21
Draw 2 different size line segments, then label the segments x and y
Duplicate each segment
Construct a segment that is 2x + y
Construct a segment that’s length is the difference of the 2 original segments
22
Duplicate segment AB using construction steps
A
B
C
D
Create a segment 2AB
Create a Segment 3CD - AB
23
In your own words, explain each of the angle theorems or postulates. Draw an example
to represent each theorem or postulate:
Right Angle Congruence Theorem:
Congruent Supplements Theorem:
Congruent Complements Theorem:
Linear Pair Postulate:
Vertical Angles Theorem:
24
Geometry 1
2.6: Proving Statements about angles
Properties of Angle
Congruence
Reflexive
Unit 2: Reasoning and Proof
Symmetric
Transitive
Right Angle Congruence
Congruent Supplements
Congruent Complements
Linear Pair Postulate
Vertical Angles Theorem
25
Investigating Complementary Angles Activity
McDougall Littell Geometry Page 108
26
Example 1
Given:
Prove:
Statement
Reason
1.
1.
2.
2.
3.
3.
4.
4.
Example 2
Given:
Prove:
Statement
Reason
1.
1.
2.
2.
3.
3.
4.
4.
27
To close a pair of scissors, you close the handles. Will the angle formed by the
blades be the same as the angle formed by the handles? Explain.
28
Example 3
Given:
Prove:
Statement
Reason
1.
1.
2.
2.
3.
3.
4.
4.
Example 4
Given:
Prove:
Statement
Reason
1.
1.
2.
2.
3.
3.
4.
4.
29
Make a sketch using the given information. Then state all pairs of congruent
angles.
Two lines intersect to form angles 1, 2, 3, 4. 1 and 2 are vertical angles. 3
and 4 are vertical and supplementary angles.
2. Solve for each variable.
(4b + 43)º
(7a + 8)º
(8a – 3)º
(6b + 17)º
30
Example 5
Example 6
Given:
Prove
Statement
Reason
1.
1.
2.
2.
3.
3.
31