# Geometry unit 2 Logic and Proof - Unit Bundle - Copy

```Logic & Proof
UNIT BUNDLE!
Unit 22- Logic & Proof: Sample Unit Outline
TOPIC
HOMEWORK
DAY 1
Inductive Reasoning, Conjectures, Counterexamples
HW #1
DAY 2
Compound Statements & Truth Tables
HW #2
DAY 3
Conditional Statements & Bi-Conditional Statements
HW #3
DAY 4
Quiz 2-1
DAY 5
Venn Diagrams
HW #4
DAY 6
Deductive Reasoning
HW #5
DAY 7
Properties of Equality; Algebraic Proofs
HW #6
DAY 8
Quiz 2-2; Introduce Properties of Congruence
DAY 9
Segment Proofs
HW #7
DAY 10
Angle Proofs
HW #8
DAY 11
Quiz 2-3
DAY 12
Unit 2 Review
Study
for Test
DAY 13
UNIT 2 TEST
None
None
None
None
See sample images of the pages on the next page.
© Gina Wilson (All Things Algebra), 2014
Name:
Class:
Topic:
Date:
Main Ideas/Questions
Notes
Inductive
Reasoning
Conjecture
Examples: Find the next five terms of the sequences then write a conjecture.
1. 38, 31, 24, 17, ______, ______, ______, ______, ______
_
Conjecture: ___________________________________________________________________________________
_______________________________________________________________________________________________
2. 2, 5, 11, 23, ______, ______, ______, ______, ______
_
Conjecture: ___________________________________________________________________________________
_______________________________________________________________________________________________
3. 1, 4, 9, 16, ______, ______, ______, ______, ______
_
Conjecture: ___________________________________________________________________________________
_______________________________________________________________________________________________
4. A, D, G, J, ______, ______, ______, ______, ______
Conjecture: ___________________________________________________________________________________
_______________________________________________________________________________________________
5. 7:30, 7:55, 8:20, _________, _________, _________, _________, _________
_
Conjecture: ___________________________________________________________________________________
_______________________________________________________________________________________________
6. 3, 1, 4, 1, 5, ______, ______, ______, ______, ______
Conjecture: ___________________________________________________________________________________
_______________________________________________________________________________________________
© Gina Wilson (All Things Algebra), 2014
Counterexample
Examples: Determine whether the conjecture is true or false. If false, provide a counterexample.
1. The sum of any two consecutive integers is always odd.
2. The product of two numbers is always larger than either number.
3. The product of two perfect squares is always a perfect square.
4. If the area of a rectangle is 6 m2, then the dimensions must be 2 meters by 3 meters.
5. Dividing by 2 always produces a number less than the original number.
6. Vertical angles are never complementary angles.
7. If a ⋅ b = 0, then either a = 0 or b = 0.
8. Two angles supplementary to the same angle must be congruent.
9. All state names have at least two syllables.
10. Squaring a number and adding one will always produce an even number.
conjecture is true or false. If false, provide a counterexample.
11. Conjecture: _______________________________________________________________________________
__________________________________________________________________________________________
T/F: ______________________________________________________________________________________
12. Conjecture: _______________________________________________________________________________
__________________________________________________________________________________________
T/F: ______________________________________________________________________________________
© Gina Wilson (All Things Algebra), 2014
Name: ___________________________________
Unit 2: Logic & Proof
Date: ________________________ Bell: ______
Homework 1: Inductive Reasoning
** This is a 2-page document! **
Directions: Find the next five terms, then write a conjecture.
1. 13, 18, 23, 28, _______, _______, _______, _______, _______
Conjecture: ___________________________________________________________________
______________________________________________________________________________
2. 512, 256, 128, 64, _______, _______, _______, _______, _______
Conjecture: ___________________________________________________________________
______________________________________________________________________________
3. 1, 8, 27, 64, _______, _______, _______, _______, _______
Conjecture: ___________________________________________________________________
______________________________________________________________________________
4. 2, 3, 5, 7, 11, 13, _______, _______, _______, _______, _______
Conjecture: ___________________________________________________________________
______________________________________________________________________________
5. I, II, III, IV, V, _______, _______, _______, _______, _______
Conjecture: ___________________________________________________________________
______________________________________________________________________________
6. 1, 1, 2, 3, 5, 8, _______, _______, _______, _______, _______
Conjecture: ___________________________________________________________________
______________________________________________________________________________
7.
,
,
, _______, _______, _______, _______, _______
Conjecture: ___________________________________________________________________
______________________________________________________________________________
© Gina Wilson (All Things Algebra), 2014
Directions: Determine if each conjecture is true or false. If false, provide a counterexample.
8. The sum of any two consecutive prime numbers is also prime.
9. The product of two even numbers is always divisible by 4.
10. The difference between two negative numbers is always negative.
11. For any integer x, x2 – x will always produce an even value.
12. For any two integers x and y, |x + y| = |x| + |y|
13. If LM = MP, then M must be the midpoint of LP.
14. All birds can fly.
15. The difference between consecutive perfect square numbers is always odd.
16. For any two integers a and b, (a + b)2 = a2 + b2
© Gina Wilson (All Things Algebra), 2014
Name:
Class:
Topic:
Date:
Main Ideas/Questions
Statement
Notes
•
A sentence that is either ____________ or ____________.
•
This is called the _____________ ________________.
•
Represented using letters such as ______ or ______.
Example
p: Supplementary angles have a sum of 180°.
Truth Value: ______
Negation
•
A negation of a statement has the ___________________ truth value.
•
Shown by the symbol ______.
Example
~p: Supplementary angles do not have a sum of 180°.
Truth Value: ______
Compound
Statements
Conjunction
Disjunction
•
Statements joined by the word __________.
•
Written as p _____ q.
•
True when ___________ statements are _____________.
•
Statements joined by the word __________.
•
Written as p _____ q.
•
True when ______ ____________ _______ statement is ____________.
Examples: Given the statements below, write compound statements. Then, determine their truth value.
p: There are seven days in a week.
q: March has 30 days.
r: Halloween is on October 31st.
1. p ∧ q: ___________________________________________________________________________________________
______________________________________________________________________ Truth Value: ________
2. q ∧ r: ___________________________________________________________________________________________
______________________________________________________________________ Truth Value: ________
3. p ∧ r: ___________________________________________________________________________________________
______________________________________________________________________ Truth Value: ________
© Gina Wilson (All Things Algebra), 2014
4. ~p ∧ q: ___________________________________________________________________________________________
______________________________________________________________________ Truth Value: ________
5. p ∨ r: ___________________________________________________________________________________________
_____________________________________________________________________ Truth Value: ________
6. ~q ∨ r: ___________________________________________________________________________________________
_____________________________________________________________________ Truth Value: ________
7. ~p ∨ ~r: __________________________________________________________________________________________
_____________________________________________________________________ Truth Value: ________
8. ~p ∨ ~q: __________________________________________________________________________________________
___________________________________________________________________ Truth Value: ________
Write
p: ______________________________________________________________________________________
q: ______________________________________________________________________________________
r: ______________________________________________________________________________________
9. p ∨ r: ___________________________________________________________________________________________
______________________________________________________________________ Truth Value: ________
10. ~p ∨ q: ___________________________________________________________________________________________
______________________________________________________________________ Truth Value: ________
11. ~q ∧ r: ___________________________________________________________________________________________
______________________________________________________________________ Truth Value: ________
12. p ∧ ~r: ___________________________________________________________________________________________
______________________________________________________________________ Truth Value: ________
13. ~p ∨ ~q: _________________________________________________________________________________________
___________________________________________________________________ Truth Value: ________
14. ~r ∧ q: _________________________________________________________________________________________
___________________________________________________________________ Truth Value: ________
© Gina Wilson (All Things Algebra), 2014
Truth Tables
Truth tables are a convenient way of organizing truth values of statements.
Complete the following negation, conjunction, and disjunction truth tables:
Negation
p
~p
T
F
p
T
T
F
F
Conjunction
q
p∧q
T
F
T
F
p
T
T
F
F
Disjunction
q
p∨q
T
F
T
F
Tips for constructing truth tables:
Include columns for each statement involved. (p, q, r, etc.)
Include columns for any negations required. (~p, ~q, ~r, etc.).
Lastly, include columns for the compound statement(s).
You try! Construct truth tables for the following compound statements:
❶ p ∨ ~r
❷ ~p ∧ q
❸ ~q ∨ r
❹ ~p ∧ ~r
© Gina Wilson (All Things Algebra), 2014
❺ ~ p ∨ ~q
❻ p ∨ (q ∧ r)
❼ ~q ∧ (p ∧ r)
❽ (~p ∨ q) ∧ ~r
© Gina Wilson (All Things Algebra), 2014
Name: ___________________________________
Unit 2: Logic & Proof
Date: ________________________ Bell: ______
Homework 2: Compound Statements
** This is a 2-page document! **
Directions: Use the statements below along with the diagram to write compound statements.
Then find its truth value.
p:
q:
r:
s:
Points C, E, and B are collinear.
∠AEC ≅ ∠DEB
EF is the angle bisector of ∠AED
∠BEC is an acute angle.
F
A
D
E
C
B
1. p ∨ q: _________________________________________________________________________________________
___________________________________________________________________ Truth Value: _________
2. q ∧ s: _________________________________________________________________________________________
____________________________________________________________________ Truth Value: ________
3. ~p ∧ r: ________________________________________________________________________________________
_________________________________________________________________ Truth Value: ________
4. r ∨ ~s: ________________________________________________________________________________________
_________________________________________________________________ Truth Value: ________
5. ~q ∧ ~r: _____________________________________________________________________________________
_______________________________________________________________ Truth Value: ________
6. p ∨ ~q: _______________________________________________________________________________________
_________________________________________________________________ Truth Value: ________
7. ~r ∨ ~s: ______________________________________________________________________________________
_________________________________________________________________ Truth Value: ________
8. ~q ∧ s: _______________________________________________________________________________________
_________________________________________________________________ Truth Value: ________
9. p ∨ r: _______________________________________________________________________________________
_________________________________________________________________ Truth Value: ________
© Gina Wilson (All Things Algebra), 2014
Directions: Complete each truth table.
10. p ∧ r
11. q ∨ s
12. ~q ∨ r
13. ~p ∧ ~s
14. r ∧ (~p ∨ q)
15. (p ∨ ~r) ∧ ~q
© Gina Wilson (All Things Algebra), 2014
Name:
Class:
Topic:
Date:
Main Ideas/Questions
Notes
•
A statement that can be written in _____-________ form.
Symbolic Form: _________________
Conditional
Statement
•
Read as “if p, then q”
or, “p ______________ q”.
The ___________________ is the phrase immediately following the word
_____.
•
The ___________________ is the phrase immediately following the word
__________.
Examples
Identify the hypothesis and conclusion of the following conditional statements:
1. If you live in Nashville, then you live in Tennessee.
Hypothesis: __________________________________________________________
Conclusion: _________________________________________________________
2. If the sum of the measures of two angles is 90°, then they are
complementary angles.
Hypothesis: __________________________________________________________
Conclusion: _________________________________________________________
3. If a quadrilateral is a square, then it has four right angles.
Hypothesis: __________________________________________________________
Conclusion: __________________________________________________________
Writing Conditional Statements:
tatements: Write the following statements in if-then form.
4. An obtuse angle has a measure greater than 90°.
__________________________________________________________________________________________________
5. All numbers divisible by 4 are also divisible by 2.
__________________________________________________________________________________________________
6. States on the east coast border the Atlantic Ocean.
__________________________________________________________________________________________________
7. Valentine’s Day is in February.
__________________________________________________________________________________________________
8. Prime numbers only have two factors, 1 and itself.
__________________________________________________________________________________________________
© Gina Wilson (All Things Algebra), 2014
Related Conditionals:
Formed by _____________________ the hypothesis and conclusion.
Inverse
Symbolic form: ___________________
Formed by _____________________ the hypothesis and conclusion.
Converse
Contrapositive
Symbolic form: ___________________
Formed by _____________________ and _____________________ the
hypothesis and conclusion.
Symbolic form: ___________________
Directions: Write the inverse, converse, and contrapositive of the following conditional statements.
Determine the truth value. If false, provide a counterexample.
1. If it is Saturday, then there is no school.
•
Inverse: _____________________________________________________________________________________
Truth Value: _______________________________________________
•
Converse: ___________________________________________________________________________________
Truth Value: _______________________________________________
•
Contrapositive: ______________________________________________________________________________
Truth Value: _______________________________________________
2. If the product of two numbers is odd, then both numbers must be odd.
•
Inverse: _____________________________________________________________________________________
____________________________________________ Truth Value: ____________________________________
•
Converse: ___________________________________________________________________________________
____________________________________________ Truth Value: ____________________________________
•
Contrapositive: ______________________________________________________________________________
____________________________________________Truth Value: _____________________________________
3. If the temperature is 25°F, then it is below freezing.
•
Inverse: _____________________________________________________________________________________
____________________________________________ Truth Value: ____________________________________
•
Converse: ___________________________________________________________________________________
____________________________________________ Truth Value: ____________________________________
•
Contrapositive: ______________________________________________________________________________
____________________________________________ Truth Value: ____________________________________
© Gina Wilson (All Things Algebra), 2014
BI-CONDITIONAL Statements
Definition:
____________________________________________________________________
Symbolic Form:
(p → q) ∧ (q → p): ______________
Read as “p if and only if q”
Bi-Conditional statements are true when __________
___________________ and _______________ are __________!
Examples: Given the bi-conditional statement below, write both the conditional and converse.
Determine the truth value of the bi-conditional. Explain why or why not.
1. Two angles are supplementary if and only if the sum of their measures is 180°.
Conditional : __________________________________________________________________
__________________________________________________________________
Converse : __________________________________________________________________
__________________________________________________________________
Truth Value? __________________________________________________________________
2. I wear my snow boots if and only if it snows.
Conditional: ___________________________________________________________________
Converse: ___________________________________________________________________
Truth Value? __________________________________________________________________
3. x2 = 25 if and only if x = 5.
Conditional: ___________________________________________________________________
Converse: ___________________________________________________________________
Truth Value? __________________________________________________________________
4. I will get 10% off if and only if I spent at least \$75.
Conditional: ___________________________________________________________________
Converse: ___________________________________________________________________
Truth Value? __________________________________________________________________
© Gina Wilson (All Things Algebra), 2014
{A graphic organizer for logic statements}
statements}
Compound Statements
Directions: Use the statements below to write the compound statements below.
p: Vertical angles are congruent.
q: 15 is a prime number.
Conjunction
____________________________________________________
________________________________Truth value: ______________
Disjunction
____________________________________________________
________________________________Truth value: ______________
Conditional Statements
Directions: Use the statements below to write the conditional statements below.
p: If it is St. Patrick’s Day.
q: It is March.
Conditional
____________________________________________________
______________________________Truth value: ________________
Inverse
____________________________________________________
______________________________Truth value: ________________
Converse
____________________________________________________
______________________________Truth value: ________________
Contrapositive
____________________________________________________
______________________________Truth value: ________________
BiBi-Conditional
____________________________________________________
______________________________Truth value: ________________
© Gina Wilson (All Things Algebra), 2014
Name: ___________________________________
Unit 2: Logic & Proof
Date: ________________________ Bell: ______
Homework 3: Conditional Statements
** This is a 2-page document! **
Directions: Identify the hypothesis and conclusion of the following conditional statements.
1. If the product of two numbers is 0, then at least one of the numbers must be 0.
Hypothesis: ______________________________________________________________________
Conclusion: ______________________________________________________________________
2. If it is daylight saving time, then I must reset my clocks.
Hypothesis: ______________________________________________________________________
Conclusion: ______________________________________________________________________
Directions: Write the following statements in if-then form.
3. A rhombus is a quadrilateral with four congruent sides. _____________________________________
_________________________________________________________________________________
4. Those that finish the marathon will get a medal. ___________________________________________
__________________________________________________________________________________
5. All freshman are required to attend orientation.___________________________________________
_________________________________________________________________________________
Directions: Write the inverse, converse, contrapositive, and bi-conditional of the conditional statements
below. Determine their truth value. If false, explain or give a counterexample.
6. If you live in Dallas, then you live in Texas.
•
Inverse: ______________________________________________________________________
_________________________________ Truth Value: ________________________________
•
Converse: ____________________________________________________________________
_________________________________ Truth Value: ________________________________
•
Contrapositive: ________________________________________________________________
_________________________________ Truth Value: ________________________________
•
Bi-Conditional: ________________________________________________________________
_________________________________ Truth Value: ________________________________
© Gina Wilson (All Things Algebra), 2014
7. If a number is a natural number, then it is also a whole number.
Inverse: ______________________________________________________________________
•
______________________________________ Truth Value: ___________________________
Converse: ____________________________________________________________________
•
______________________________________ Truth Value: ___________________________
Contrapositive: ________________________________________________________________
•
______________________________________ Truth Value: ___________________________
Bi-Conditional: ________________________________________________________________
•
_______________________________________ Truth Value: ___________________________
Directions: Write the appropriate statement to match the symbolic notation. Then, classify it as the
conditional, inverse, converse, contrapositive, or bi-conditional.
p: you have a library card
q: you can check out books
8. p → q: __________________________________________________________________________
____________________________________ Classify: ____________________________
9. ~q → ~p: _______________________________________________________________________
_________________________________ Classify: ____________________________
10. q → p: _________________________________________________________________________
__________________________________ Classify: ____________________________
11. ~p → ~q: _______________________________________________________________________
_________________________________ Classify: ____________________________
12. p
↔ q:
________________________________________________________________________
_________________________________ Classify: ____________________________
© Gina Wilson (All Things Algebra), 2014
Name: ________________________________________
Geometry
Date: ______________________________ Per: ______
Unit 2: Logic & Proof
Quiz 2-1: Conjectures, Compounds, and Conditionals
Determine if the conjectures below are true or false. If false, give a counterexample.
1. The difference of two odd numbers is always an odd number.
__________________________________________________________________________________
2. Two right angles are always supplementary.
_________________________________________________________________________________
3. For any two integers a and b, (a – b)2 = a2 – b2
_________________________________________________________________________________
Match each statement with its symbolic notation.
_________ 4. Conjunction
A. q → p
_________ 5. Disjunction
B. p
_________ 6. Conditional
C. p ∨ q
_________ 7. Inverse
D. ~p → ~q
_________ 8. Converse
E. ~q → ~p
_________ 9. Contrapositive
F. p ∧ q
_________10. Bi-Conditional
G. p → q
↔q
Use the statements below to write compound statements. Then, determine the truth value.
p: Atlanta is the capital of Georgia.
q: There are 4 feet in a yard.
11. p ∨ q: ________________________________________________________________________________________
_________________________________________________________________ Truth Value: _________
12. p ∧ q: ________________________________________________________________________________________
__________________________________________________________________ Truth Value: _________
13. p ∧ ~q: ______________________________________________________________________________________
________________________________________________________________ Truth Value: _________
14. ~p ∨ q: ______________________________________________________________________________________
_______________________________________________________________ Truth Value: _________
© Gina Wilson (All Things Algebra), 2014
Complete the truth tables below.
15. ~p ∨ ~q
16. p ∧ ~r
Use the conditional statement “If it is January, then there is snow.” for questions 17 – 18.
17. Identify the hypothesis: _____________________________________________________________
18. Identify the conclusion: _____________________________________________________________
Write the inverse, converse, contrapositive, and bi-conditional of the conditional statement
below. Determine their truth value. If false, explain or give a counterexample.
19. If two angles form a linear pair, then they are supplementary.
Inverse: _______________________________________________________________________
________________________________________________________________________
Truth Value: ______________________________________________________________
Converse: _______________________________________________________________________
_______________________________________________________________________
Truth Value: _____________________________________________________________
Contrapositive: _____________________________________________________________________
_____________________________________________________________________
Truth Value: ___________________________________________________________
20. Write the conditional and converse of the bi-conditional below. Then, determine its truth value.
“A capital letter is a vowel if and only if it is symmetrical.”
Conditional: _____________________________________________________________________
Converse: ______________________________________________________________________
Truth Value of Bi-Conditional? Explain. ___________________________________________
_________________________________________________________________________________
© Gina Wilson (All Things Algebra), 2014
VENN DIAGRAMS
Venn Diagrams are a visual way of displaying the relationships between sets.
Types of Venn Diagrams
All, Always, Every
Some, Sometimes
Never, No, None
q
p
q
q
p
p
All elements of p
are elements of q.
Some elements of p
are elements of q.
There is no relationship
between p and q.
Directions: Draw a Venn Diagram to represent each statement.
1. Some students who take chorus also
2. No perfect squares are prime numbers.
take band.
3. Numbers divisible by 6 are always
divisible by 3.
4. Every natural number is a whole number.
5. Some vertical angles are
complementary.
6. Irrational numbers are never rational
numbers.
Directions: Shade the indicated region on the Venn Diagram.
7. q
8. p ∧ q
p
q
p
q
© Gina Wilson (All Things Algebra), 2014
9. p ∨ q
10. p ∧ ~q
p
q
p
q
Directions: Describe each diagram using a conditional or compound statement.
11.
Acute
_______________________________________________________
Angles
_______________________________________________________
Angles that
measure 60°°
_______________________________________________________
12.
_______________________________________________________
Dogs
_______________________________________________________
_______________________________________________________
Beagles
13.
_______________________________________________________
_______________________________________________________
_______________________________________________________
Volleyball
Piano Lessons
14.
_______________________________________________________
_______________________________________________________
Months
with 31 days
Months that
_______________________________________________________
15.
_______________________________________________________
_______________________________________________________
_______________________________________________________
Art Class
Freshman
16.
_______________________________________________________
_______________________________________________________
_______________________________________________________
Multiples of 7
Even Numbers
© Gina Wilson (All Things Algebra), 2014
More Practice with Venn Diagrams
1. The Venn diagram below shows the
number of senior boys who play
football or wrestle.
Wrestling
Football
18
104
2. The Venn diagram below shows the
number of people at the gym signed
up for yoga or aerobics class.
Aerobics
Yoga
24
68
b. How many senior boys play football but do not
wrestle? _______
c. How many senior boys play football or
wrestle? _______
12
9
a. How many senior boys play football and
wrestle? _______
d. How many senior boys wrestle? _______
a. How many people are signed up for yoga
class? _______
b. How many people are signed up for yoga or
aerobics? _______
c. How many people are signed up for aerobics
but not yoga? _______
89
575
d. How many people belong to the gym? _______
3. The Venn diagram below shows
the number of 8th grade students
who have a smartphone or tablet.
Smartphone
Tablet
a. How many students have a smartphone and a
tablet? _______
b. How many students do not have a smartphone
or tablet? _______
c. How many 8th grade students are there? ______
58
116
29
d. How many students have a tablet? _______
117
e. How many students have a smartphone but not
a tablet? _______
4. The Venn diagram below shows
the number of sophomores with
accounts.
37
124
46
a. How many sophomores just have a Facebook
account? _______
b. How many sophomores have a Twitter or
Instagram account? _______
and Instagram accounts? _______
21
d. How many students do not have a Twitter
account? ______
16
55
34
28
Instagram
e. How many students do not have a Facebook
and Instagram account? _______
© Gina Wilson (All Things Algebra), 2014
5. The Venn diagram below shows survey results of cities visited by group of people on a
recent trip to Europe.
London
Paris
64
95
89
24
43
31
76
Dublin
78
a. How many people visited Dublin or London? ________
b. How many people visited London and Paris? ________
c. How many people visited London or Paris? ________
d. How many people visited Dublin and not London?________
e. How many people only visited Dublin? ________
f. How many people visited Dublin and Paris? ________
g. How many people visited all three cities? ________
h. How many people visited London? ________
i. How many people took the survey? ________
6. On a recent field trip to Cedar Point, students reported which rollercoasters they went on.
The Venn Diagram below represents the results of the survey.
Blue Streak
Maverick
10
24
27
14
21
19
38
12
Millennium
Force
a. How many students just went on the Maverick? ________
b. How many students went on the Blue Streak or the Millennium Force? ________
c. How many students went on the Millennium Force and Maverick? ________
d. How many students went on the Blue Streak?________
e. How many students went on the Blue Streak and Maverick, but not the Millennium
Force? ________
© Gina Wilson (All Things Algebra), 2014
Name: ___________________________________
Unit 2: Logic & Proof
Date: ________________________ Bell: ______
Homework 4: Venn Diagrams
** This is a 2-page document! **
Directions: Draw a Venn diagram to represent each statement.
1. Trapezoids are never parallelograms.
2. Every apple is a fruit.
3. All linear pairs are supplementary angles.
4. Some teens who babysit also mow lawns.
Directions: Shade the indicated region of the Venn diagrams below.
5. p
6. p ∧ q
7. p ∨ q
p
q
p
8. Some students who take band also take art. If Jack
takes art but not band, shade the area on the diagram
where he would belong.
9. There are three classes offered at the craft store:
photography, cake decorating, and scrapbooking. If
Sarah is signed up for photography and scrapbooking,
but not cake decorating, shade the area on the diagram
where she would belong.
p
q
Band
q
Art
Photography
Cake Decorating
Scrapbooking
Directions: Describe each Venn diagram using a conditional or compound statement.
10.
Rectangles
____________________________________________________________
Squares
____________________________________________________________
© Gina Wilson (All Things Algebra), 2014
11.
Military
____________________________________________________________
____________________________________________________________
Marines
12.
____________________________________________________________
____________________________________________________________
Odd Numbers
Perfect Squares
13.
____________________________________________________________
____________________________________________________________
Boy Scouts
Baseball Players
14.
____________________________________________________________
____________________________________________________________
Letters in
Mississippi
Vowels
15. On a recent survey, students were asked if they ice skate, snowboard, or ski. The Venn diagram
below shows the results of the survey.
Ice Skate
Ski
2
7
13
3
4
8
10
6
a. How many students ski or snowboard? ________
b. How many students ice skate and ski? ________
c. How many students ice skate, ski, and snowboard?
________
d. How many students do not ski or ice skate? ________
e. How many students just snowboard? ________
Snowboard
f. How many students do not ice skate, ski, or
snowboard? ________
g. How many students ice skate? ________
h. How many students do not snowboard?________
i. How many students took the survey? ________
© Gina Wilson (All Things Algebra), 2014
Name:
Class:
Topic:
Date:
Main Ideas/Questions
Notes
Deductive
Reasoning
Given a conditional statement, if the __________________ is
_________, then the ___________________ is __________.
Law of
Detachment
Examples:
Symbolic Map:
Use the Law of Detachment to give a valid conclusion.
If not possible, write no valid conclusion.
1. Given: If Mark earns saves \$30, he can buy a new video game.
Mark saves \$30.
_
Conclusion: _______________________________________________________________________________
2. Given: If a quadrilateral is a rhombus, then it is also a parallelogram.
_
Conclusion: _______________________________________________________________________________
3. Given: If you are 18 years old, then you can register to vote.
Olivia is not 18 years old.
_
Conclusion: _______________________________________________________________________________
4. Given: If two the sum of the measures of two angles is 90°, then they are complementary.
m∠J = 58° and m∠K = 32°
_
Conclusion: _______________________________________________________________________________
5. Given: If you plan to attend prom, then you must purchase a ticket.
Sarah purchases a prom ticket.
_
Conclusion: _______________________________________________________________________________
© Gina Wilson (All Things Algebra), 2014
Allows you to draw a conclusion from ______ conditional
statements in which the__________________ of the first
Law of
statement is the __________________ of the second statement.
Syllogism
Symbolic Map:
Examples:
Use the Law of Syllogism to give a valid conclusion.
If not possible, write no valid conclusion.
6. Given: If it is Saturday, then Jake has a baseball tournament.
If Jake has a baseball tournament, then he will need to pack his lunch.
_
Conclusion: ________________________________________________________________________________
_________________________________________________________________________________
7. Given: If a number is divisible by 12, then it is divisible by 6.
If a number is divisible by 6, then it is divisible by 3.
_
Conclusion: ________________________________________________________________________________
_________________________________________________________________________________
8. Given: If a quadrilateral is a square, then it is a rectangle.
If a quadrilateral is a rectangle, then it has four right angles.
Conclusion: ________________________________________________________________________________
_________________________________________________________________________________
9. Given: If it is sunny this weekend, then you will go boating.
If it is sunny this weekend, then you will wear shorts.
Conclusion: ________________________________________________________________________________
_________________________________________________________________________________
10. Given: If you shop at Target, then you will use your Target Red Card.
If you do not use your Target Red Card, then you will not get 5% off.
Conclusion: ________________________________________________________________________________
_________________________________________________________________________________
11. Given: If it snows, then school will be canceled.
If school is canceled, then students will need to make-up a day of school.
Conclusion: ________________________________________________________________________________
_________________________________________________________________________________
© Gina Wilson (All Things Algebra), 2014
More Practice:
Practice: Law of Detachment & Law of Syllogism
Law of Detachment
Law of Syllogism
p→q
p
∴q
p→q
q→r
∴p→r
Determine whether the conclusion follows from the given statements
by the Law of Detachment or the Law of Syllogism. If it does, state which
law was used. If not, write invalid.
1. Given: If I drive over the speed limit, then I will get a ticket.
Given: I drove over the speed limit.
Conclusion: I got a ticket.
2. Given: If Amanda goes to the restaurant, then she will order a hamburger.
Given: If she orders a hamburger, then she will get fries.
Conclusion: If Amanda goes to the restaurant, then she will get fries.
3. Given: If it snows, then I will wear my snow boots.
Given: If it snows, then I won’t go to school that day.
Conclusion: If I don’t go to school, then I will wear my snow boots.
4. Given: If Tina goes to the beach, she will wear sunscreen.
Given: Tina goes to the beach.
Conclusion: Tina will wear sunscreen.
5. Given: If you get a cold, then you eat chicken soup.
Given: You ate chicken soup.
Conclusion: You must have a cold.
© Gina Wilson (All Things Algebra), 2014
6. Given: If two angles form a linear pair, then they are supplementary.
Given: If two angles are supplementary, then the sum of their measures is 180°.
Conclusion: If two angles form a linear pair, then the sum of their measures is 180°.
7. Given: If Kaylee gets at least a 95 on her final exam, she will get an A in Geometry.
Given: Kaylee got a 98 on her final exam.
Conclusion: Kaylee got an A in Geometry.
8. Given: If it is raining, then you must bring an umbrella.
Conclusion: It is raining.
9. Use the Law of Syllogism to write three conditional statements that can be formed from the
following true statements.
a. If a quadrilateral is a square, then it has four right angles.
b. If a quadrilateral is a rhombus, then its opposite sides are parallel.
c. If a quadrilateral has four right angles, then it is a rectangle.
d. If a quadrilateral is a rectangle, then its opposite sides are congruent.
e. If the opposite sides of a quadrilateral are parallel, then it is a parallelogram.
____________________________________________________________________
____________________________________________________________________
____________________________________________________________________
© Gina Wilson (All Things Algebra), 2014
Name: ___________________________________
Unit 2: Logic & Proof
Date: ________________________ Bell: ______
Homework 5: Deductive Reasoning
** This is a 2-page document! **
Directions: Use the Law of Detachment to give a valid conclusion.
If not possible, write no valid conclusion.
1. Given: If Logan eats his vegetables, then he can have a bowl of ice cream.
Logan eats his vegetables.
Conclusion: _______________________________________________________________________
2. Given: If a polynomial is prime, then it cannot be factored.
5x + 13y is prime.
Conclusion: _______________________________________________________________________
3. Given: If you visit Paris, then you will see the Eiffel Tower.
You did not see the Eiffel Tower.
Conclusion: _______________________________________________________________________
4. Given: If the measure of an angle is greater than 90°, then it is an obtuse angle.
m∠PQR = 115°
Conclusion: _______________________________________________________________________
Directions: Use the Law of Syllogism to give a valid conclusion.
If not possible, write no valid conclusion.
5. Given: If Nicole is tardy to class again, she will get detention.
If Nicole gets detention, her mother will take away her phone.
Conclusion: ______________________________________________________________________
______________________________________________________________________
6. Given: If it is raining, then I will bring my umbrella.
If I do not bring my umbrella, then I do not need my rain boots.
Conclusion: ______________________________________________________________________
______________________________________________________________________
7. Given: If natural number ends in 0, then it is divisible by 10.
If a natural number is divisible by 10, then it is divisible by 5.
Conclusion: ______________________________________________________________________
______________________________________________________________________
© Gina Wilson (All Things Algebra), 2014
Directions: Determine whether the conclusion follows from the given statements by the Law of
Detachment of the Law of Syllogism. If it does, state which law was used.
If not, write invalid.
8. Given: If you go camping, then you will need a flashlight.
Given: If you need a flashlight, then you will need extra batteries.
Conclusion: If you go camping, then you will need extra batteries.
9. Given: If Sally goes to the airport, then she will need to pay for parking.
Given: If Sally does not pay for parking, then her car will get towed.
Conclusion: If Sally does not go to the airport, then her car will not get towed.
10. Given: If you leave the country, then you will need a passport.
Given: Nate is planning a trip to Italy.
Conclusion: Nate will need a passport.
11. Given: If you buy one pair of shoes, then you get another pair for 50% off.
Given: Carolyn does not buy one pair of shoes.
Conclusion: Carolyn does not get another pair for 50% off.
12. Given: If the sum of the measures of two angles is 180°, then they are supplementary.
Given: m∠D = 110° and m∠E = 70°
Conclusion: ∠D and ∠E are supplementary angles.
13. Given: If a number is a multiple of 16, then it is a multiple of 8.
Given: If a number is a multiple of 8, then it is a multiple of 4.
Conclusion: If a number is a multiple of 16, then it is a multiple of 4.
© Gina Wilson (All Things Algebra), 2014
Properties of Equality
Property
Notes
of Equality
If ___________, then ________________________.
Subtraction Property
of Equality
If ___________, then ________________________.
Multiplication Property
of Equality
If ___________, then ________________________.
Division Property
of Equality
If ___________, then ________________________.
Distributive
Property
If a(b + c), then a(b + c) = ______________________.
Substitution
Property
If ____________ then a may be _________________
by b in any expression or equation.
Reflexive
Property
For any real number a, _______________.
Symmetric
Property
If _____________, then _____________.
Transitive
Property
If ___________ and___________, then ___________.
(A value always will equal itself!)
Using the Properties of Equality
Properties of equality can be used to justify steps in solving an equation.
NEW!
TwoTwo-Column Proof: A common format used to organize a proof.
Left Side: List the _______________ (or steps).
Right Side: List the _______________ that justify each step.
What can be used as reasons?
_________________, _________________, _________________, _________________,
❶ Given: 4x – 1 = 27; Prove: x = 7
1. 4x – 1 = 27
1. Given
2. 4x = 28
2.
3. x = 7
3.
© Gina Wilson (All Things Algebra), 2014
❷ Given:
a
+ 2 = 5 ; Prove:
−6
a = -18
a
+2=5
−6
a
=3
2.
−6
1.
3. a = -18
1. Given
2.
3.
❸ Given: -9(2x – 3) = 63; Prove: x = -2
1. -9(2x – 3) = 63
1. Given
2. -18x + 27 = 63
2.
3. -18x = 36
3.
4. x = -2
4.
❹ Given: 6x + 7 = 8x – 17; Prove: x = 12
1. 6x + 7 = 8x – 17
1. Given
2. 7 = 2x – 17
2.
3. 24 = 2x
3.
4. 12 = x
4.
5. x = 12
5.
❺ Given: -7(x + 2) + 4x = 6(2x – 4); Prove: x = 1.5
1. -7(x + 2) + 4x = 6(2x – 4)
1. Given
2. -7x – 14 + 4x = 12x – 24
2.
3. -3x – 14 = 12x – 24
3.
4. -15x – 14 = -24
4.
5. -15x = -10
5.
6. x = 1.5
6.
© Gina Wilson (All Things Algebra), 2014
Algebraic Proofs
Proofs
❶ Given: 3x + 1 = -14; Prove: x = -5
Statements
Reasons
❷ Given: 2(x – 9) = -10; Prove: x = 4
Statements
❸ Given:
m
+ 10 = −1 ; Prove: m = 33
−3
Statements
❹ Given:
Reasons
Reasons
5y − 1
= 7 ; Prove: y = 3
2
Statements
Reasons
© Gina Wilson (All Things Algebra), 2014
❺ Given: 10k – 4 = 2k – 20; Prove: k = -2
Statements
Reasons
❻ Given: 5n – 42 = 12n; Prove: n = -6
Statements
Reasons
❼ Given: -8(w + 1) = -5(w + 10); Prove: w = 14
Statements
Reasons
❽ 14 – 2(x + 8) = 5x – (3x – 34); Prove: x = -9
Statements
Reasons
© Gina Wilson (All Things Algebra), 2014
Name that Property!
Directions: Match the statements with the properties of equality.
Properties will be used more than once.
_______ 1. If k = 3, then 3 = k
B. Subtraction Property of Equality
_______2.
2. If 2x = 14, then x = 7
C. Multiplication Property of Equality
_______3
3. 4 = 4
E. Distributive Property
_______4.
4. If -5x – 1 = -11, then -5x = -10
_______5
5. If 10a = 2b and 2b = c, then 10a = c
D. Division Property of Equality
F. Substitution Property
G. Reflexive Property
H. Symmetric Property
I. Transitive Property
_______6
6. -7(x – 4) = -7x + 28
_______77. If 6y = 24, then 6y – 3 = 24 – 3
_______8
8. If 10x + w = 41 and w = 1, then 10x + 1 = 41
_______9
9. If
y
= −10 , then y = -20
2
_______10
10.
10. If 3x = 2y and 2y = z, then 3x = z
_______1111.
11. If 7m = 35, then 7m + 4 = 35 + 4
_______12
12.
12. If -2x = 18, then 18 = -2x
_______13
13.
13. Given 3x2 + 1, if x = 5, then 3(5)2 + 1
_______14
14.
14. If m = -2, then 8m = -16
_______15
15.
15. 10y = 10y
_______16
16.
16. 5x + 8x = x(5 + 8)
© Gina Wilson (All Things Algebra), 2014
Name: ___________________________________
Unit 2: Logic & Proof
Date: ________________________ Bell: ______
Homework 6: Algebraic Proof
** This is a 2-page document! **
Directions: Name the property of equality that justifies each statement.
_______1. If a = 2b, then a – c = 2b – c
_______2. x = x
B. Subtraction Property of Equality
C. Multiplication Property of Equality
_______3. 3(p – 7) = 3p – 21
D. Division Property of Equality
_______4. If -7k = -42, then k = 6
E. Distributive Property
_______5. If m + n = 15 and n = 2, then m + 2 = 15
x
_______6. If = −5 , x = -20
4
2
F. Substitution Property
G. Reflexive Property
H. Symmetric Property
2
_______7. If w = 2x and 2x = y, then w = y
I. Transitive Property
_______8. If c – 9 = -1, then c = 8
_______9. If n = -3, then -3 = n
Directions: Complete each proof using the properties of equality. Not all rows may be used.
10. Given: -8(x – 3) = -32; Prove: x = 7
Statements
11. Given:
− 16 =
Reasons
m
− 18 ; Prove: m = 10
5
Statements
Reasons
© Gina Wilson (All Things Algebra), 2014
12. Given:
2y −1
= −5 ; Prove: y = 8
−3
Statements
Reasons
13. Given: 2x + 30 = -4(5x – 2); Prove: x = -1
Statements
Reasons
14. Given: 18x – 2(3x + 1) = 5x – 16; Prove: x = -2
Statements
Reasons
© Gina Wilson (All Things Algebra), 2014
Name: ________________________________________
Geometry
Date: ______________________________ Per: ______
Unit 2: Logic & Proof
Quiz 2-2: Venn Diagrams, Deductive Reasoning, & Algebra Proof
Draw a Venn Diagram to represent each relationship.
1. Some people check out books
2. There are no freshmen in AP
and DVDs from the library.
Economics.
3. All dimes are coins.
Shade the indicated region of the Venn diagrams below.
4. Some students who are in chorus also take
singing lessons. If Adam is in Chorus and
takes singing lessons, shade the area where
Singing
Lessons
Chorus
5. A group of student went to the Holiday Valley
Ski Resort for a field trip. There were three
black diamond slopes open: the Raven, the
Falcon, and the Eagle. Shade the region that
represents the students who went on the Falcon
or the Eagle.
Falcon
Raven
Eagle
6. The Venn diagram below shows the number of
seniors who have their driver’s license, have
part-time jobs, and are taking college credit
classes.
Driver’s
Part-Time
Jobs
101
24
60
45
7 18
32
15
College
Classes
_______a. How many seniors have part-time jobs?
_______b. How many seniors have a driver’s
_______c. How many seniors have a part-time job
or take college classes?
_______d. How many seniors take college classes
but do not have their driver’s license?
_______e. How many seniors have a driver’s
license, have a part-time job, and take
college classes?
_______f. How many seniors do not take college
classes?
_______g. How many seniors are there?
© Gina Wilson (All Things Algebra), 2014
Use the Law of Detachment or the Law of Syllogism to reach a valid conclusion. If a valid
conclusion cannot be reached, write no conclusion.
7. Given: If Alex runs a mile in under 6 minutes, then he will have a new personal record.
Given: Alex ran a mile in 5 minutes and 45 seconds.
Conclusion: ______________________________________________________________________
______________________________________________________________________
8. Given: If you go to the water park, then you will need to bring sunscreen.
Given: If you do not bring sun screen, then you will burn.
Conclusion: ______________________________________________________________________
______________________________________________________________________
9. Given: If the measures of two angles in a triangle are 30° and 60°, then the third angle is 90°.
Given: If a triangle has a 90° angle, then it is a right triangle.
Conclusion: ______________________________________________________________________
______________________________________________________________________
State whether the Law of Detachment or the Law of Syllogism was used to draw the
conclusion from the given statements. If it’s an invalid conclusion, write invalid.
10. Given: If a number is a whole number, then it is an integer.
Given: 3 is a whole number
Conclusion: 3 is an integer
11. Given: If the Patriots win their next game, then they will win their division.
Given: If the Patriots win their next game, then they will go to the playoffs.
Conclusion: If the Patriots win their division, then they will go the playoffs.
12. Given: If two lines are parallel, then they never intersect.
Given: If two lines never intersect, then they have the same slope.
Conclusion: If two lines are parallel, then they have the same slope.
13. Given: If you sell \$1000 worth of popcorn, then your Boy Scout dues are free.
Given: Ethan sold \$1000 worth of popcorn.
Conclusion: Ethan’s Boy Scout dues are free.
14. Given: If you are at least 18 years old, then you can apply for a credit card.
Given: Kelly applied for a credit card.
Conclusion: Kelly is at least 18 years old.
© Gina Wilson (All Things Algebra), 2014
Write the letter of the property that justifies each statement in the blank.
_______15. If 4m + n = 7 and n = 3, then 4m + 3 = 7
_______16. If a = -5, then 2a = -10
B. Subtraction Property of Equality
_______17. If 3x = y, then y = 3x
C. Multiplication Property of Equality
D. Division Property of Equality
_______18. If 4w – 1 = 11, then 4w = 12
_______19. If a + b = c and c = d 2, then a + b = d 2
E. Distributive Property
F. Substitution Property
_______20. 6y = 6y
G. Reflexive Property
_______21. If -8q = -56, then q = 7
H. Symmetric Property
_______22. If 5k = -25, then 5k – 1 = -25 – 1
I. Transitive Property
_______23. -3(2x – 5) = -6x + 15
Complete each proof using the properties of equality. Not all rows may be used.
24. Given:
− 5x + 2
= −4 ; Prove: x = 6
7
Statements
25. Given: 17 – 3(w + 5) = 6(w + 5) – 2w;
Statements
Reasons
Prove: w = -4
Reasons
© Gina Wilson (All Things Algebra), 2014
Segments Proofs Reference
Properties of Equality
Subtraction Property
Multiplication Property
Division Property
Distributive Property
Substitution Property
Reflexive Property
Symmetric Property
Transitive Property
The properties above may only be used with EQUAL signs. The following
properties of congruence can be applied to statements with congruence symbols:
Properties of Congruence
Reflexive Property
of Congruence
For any segment AB, ____________________.
Symmetric Property
of Congruence
If ___________________, then ___________________.
Transitive Property
of Congruence
If ____________________ and ____________________,
then ____________________.
Definitions
Segments are congruence if and only if
they have the same measure:
Definition of
Congruence
If __________________, then __________________.
If __________________, then __________________.
The midpoint of a segment divides the
segment into 2 equal (congruent) parts.
Definition of
Midpoint
If M is the midpoint of AB, then _________________
Postulates
If A, B, and C are collinear points and B is between A and C:
Postulate
A
B
C
then: _______________________________________
© Gina Wilson (All Things Algebra), 2014
Practice!
Justify each of the following statements using a property of
equality, property of congruence, definition, or postulate.
1. If PQ = PQ, then PQ ≅ PQ
___________________________________
2. If K is between J and L, then JK + KL = JL
___________________________________
3. EF ≅ EF
____________________________________
4. If RS = TU, then RS + XY = TU + XY
___________________________________
5. If AB = DE, then DE = AB
___________________________________
6. If Y is the midpoint of XZ , then XY = YZ
___________________________________
7. If FG ≅ HI and HI ≅ JK , then FG ≅ JK
___________________________________
8. If AB + CD = EF + CD, then AB = EF
___________________________________
9. If PQ + RS = TV and RS = WX, then
PQ + WX = TV
___________________________________
10. If LP = PN, and L, P, and N are collinear,
___________________________________
then P is the midpoint of LN
11. If UV ≅ UV , then UV = UV
___________________________________
12. If CD + DE = CE, then CD = CE – DE
___________________________________
Property Bank:
Properties of Equality:
Properties of Congruence:
Reflexive Property
Subtraction Property
Symmetric Property
Multiplication Property
Transitive Property
Division Property
Distributive Property
Substitution Property
Reflexive Property
Definitions:
Definition of Congruence
Definition of Midpoint
Symmetric Property
Postulates:
Transitive Property
© Gina Wilson (All Things Algebra), 2014
Segment Proofs
Proofs Guide
Segment Proofs
Proofs Guide
Directions: Use the reasons below to complete proofs 1-6.
Directions: Use the reasons below to complete proofs 1-6.
1
1
2
Given
Transitive Property
Definition of Midpoint
Simplify
3
Definition of Midpoint
Definition of Congruence
Given
Transitive Property
4
Transitive Property
Substitution Property
Definition of Congruence
Substitution Property
Definition of Congruence
Given
5
Transitive Property
Definition of Congruence
Substitution
Given
Definition of Congruence
2
Given
Transitive Property
Definition of Midpoint
Simplify
3
Given
Substitution Property
Transitive Property
Definition of Midpoint
Given
Definition of Congruence
Symmetric Property
6
© Gina Wilson (All Things Algebra), 2014
4
Transitive Property
Substitution Property
Definition of Congruence
Substitution Property
Definition of Congruence
Given
5
Definition of Congruence
Substitution
Given
Definition of congruence
Transitive Property
Definition of Midpoint
Definition of Congruence
Given
Transitive Property
Transitive Property
Definition of Congruence
Substitution
Given
Definition of Congruence
Given
Substitution Property
Transitive Property
Definition of Midpoint
Given
Definition of Congruence
Symmetric Property
6
Definition of Congruence
Substitution
Given
Definition of congruence
Transitive Property
© Gina Wilson (All Things Algebra), 2014
Segments Proofs
Directions:
1
Complete the proofs below by giving the missing statements and reasons.
Given: E is the midpoint of DF
Prove: 2DE = DF
D
Statements
E
F
Reasons
1. E is the midpoint of DF
1.
2. DE = EF
2.
3. DE + DE = DE + EF
3.
4. 2DE = DE + EF
4.
5. DE + EF = DF
5.
6. 2DE = DF
6.
K
2
Given: KL ≅ LN , LM ≅ LN
Prove: L is the midpoint of KM
L
N
M
Statements
Reasons
1. KL ≅ LN , LM ≅ LN
1.
2. KL = LN, LM = LN
2.
3. KL = LM
3.
4. L is the midpoint of KM
4.
U
P
3
Given:
PQ ≅ TQ , UQ ≅ QS
Prove:
PS ≅ TU
Q
T
Statements
S
Reasons
1. PQ ≅ TQ , UQ ≅ QS
1.
2. PQ = TQ, UQ = QS
2.
3. PQ + QS = PS; TQ + QU = TU
3.
4. TQ + QS = PS
4.
5. TQ + QS = TU
5.
6. PS = TU
6.
7. PS ≅ TU
7.
© Gina Wilson (All Things Algebra), 2014
J
4
Given: K is the midpoint of JL, M is the midpoint of LN,
JK = MN
Prove: KL ≅ LM
K
M
L
Statements
5
N
Reasons
1. K is the midpoint of JL,
M is the midpoint of LN
1.
2. JK = KL, LM = MN
2.
3. JK = MN
3.
4. MN = KL, LM = MN
4.
5. LM = KL
5.
6. KL = LM
6.
7. KL ≅ LM
7.
Given: XY ≅ UV , YZ ≅ TU
Prove: XZ ≅ TV
X
Y
T
Statements
U
Z
V
Reasons
1. XY ≅ UV , YZ ≅ TU
1.
2. XY = UV, YZ = TU
2.
3. XY + YZ = XZ, TU + UV = TV
3.
4. UV + YZ = XZ, YZ + UV = TV
4.
5. XZ = TV
5.
6. XZ ≅ TV
6.
X
6
Given: YW ≅ YZ , XY ≅ VY
Prove: XZ ≅ VW
W
Y
Z
V
Statements
Reasons
1. WY ≅ YZ , XY ≅ VY
1.
2. WY = YZ, XY = VY
2.
3. XY + YZ = XZ
3.
4. VY + YW = XZ
4.
5. VY + YW = VW
5.
6. XZ = VW
6.
7. XZ ≅ VW
7.
© Gina Wilson (All Things Algebra), 2014
A
7
Given: E is the midpoint of AC , DE = EC
Prove: DE ≅ AE
E
C
D
Statements
8
Reasons
1. E is the midpoint of AC
1.
2.
2. Definition of Midpoint
3.
3. Given
4. AE = DE
4.
5.
5. Definition of Congruence
6. DE ≅ AE
6.
Given: RS =
1
RT
2
R
S
T
Prove: S is the midpoint of RT
Statements
1. RS =
9
1
RT
2
Reasons
1.
2. 2RS = RT
2.
3.
4. 2RS = RS + ST
4.
5. RS = ST
5.
6.
6. Definition of Midpoint
Given: M is the midpoint of LN ,
N is the midpoint of MO
Prove: LM ≅ NO
L
Statements
M
N
O
Reasons
1. M is the midpoint of LN
1.
2. LM = MN
2. Definition of Midpoint
3.
3. Given
4. MN = NO
4.
5.
5. Transitive Property of Equality
6.
6. Definition of Congruence
© Gina Wilson (All Things Algebra), 2014
10
Given: Y is the midpoint of XZ
Prove:
XY =
1
XZ
2
Statements
11
Reasons
Given: AC ≅ DF , BC ≅ DE
Prove: AB ≅ EF
A
B
D
Statements
12
Given: AB ≅ CD
Prove: AC ≅ BD
Statements
C
E
F
Reasons
A
B
C
D
Reasons
© Gina Wilson (All Things Algebra), 2014
Name: ___________________________________
Unit 2: Logic & Proof
Date: ________________________ Bell: ______
Homework 7: Segment Proofs
** This is a 2-page document! **
Use the segment addition postulate to write three equations using the diagram below.
1. _______________________________________
P
Q
R
S
T
2. _______________________________________
3. _______________________________________
Complete the proofs below by filling in the missing statements and reasons
Z
4.
Given: X is the midpoint of WY , WX ≅ XZ
Prove: XY ≅ XZ
W
Statements
5.
Y
Reasons
1. X is the midpoint of WY
1.
2. WX = XY
2.
3. WX ≅ XZ
3.
4. WX = XZ
4.
5. XY = XZ
5.
6. XY ≅ XZ
6.
Given:
X
AB ≅ CD
A
Prove: AC ≅ BD
Statements
B
C
D
Reasons
1. AB ≅ CD
1.
2. AB = CD
2.
3. AC + CD = AD
3.
4. AB + BD = AD
4.
5. CD + BD = AD
5.
6. AC + CD = CD + BD
6.
7. AC = BD
7.
8. AC ≅ BD
8.
© Gina Wilson (All Things Algebra), 2014
6.
Given: 2PQ = PR
P
Prove: Q is the midpoint of PR
Statements
Q
R
Reasons
1.
1. Given
2.
3. 2PQ = PQ + QR
3.
4. PQ = QR
4.
5.
5. Definition of Midpoint
C
7.
B
A
Given: AB ≅ CD , BD ≅ DE
D
E
Statements
1.
8.
AB ≅ CD , BD ≅ DE
Reasons
1. Given
2.
2. Definition of Congruence
3. AB + BD = AD
3.
4. CD + DE = AD
4.
5.
6.
7.
7. Definition of Congruence
Given: GI ≅ JL , GH ≅ KL
Prove: HI ≅ JK
G
H
J
Statements
I
K
L
Reasons
© Gina Wilson (All Things Algebra), 2014
Angle Proofs Reference
Properties of Equality
Subtraction Property
Multiplication Property
Division Property
Distributive Property
Properties of Congruence
Substitution Property
Reflexive Property
Symmetric Property
Transitive Property
Reflexive Property
Symmetric Property
Transitive Property
Definitions
Definition of
Congruence
m∠A = m∠B ↔ ∠A ≅ ∠B
Definition of
Angle Bisector
An angle bisector divides an angle into two equal parts.
Definition of
Complementary Angles
Complementary ↔ Sum is 90°.
Definition of
Supplementary Angles
Supplementary ↔ Sum is 180°.
Definition of
Perpendicular
Perpendicular lines form right angles.
Definition of
a Right Angle
A right angle = 90°.
Postulates
Postulate
A
m∠ABD + m∠DBC = ∠ABC
D
B
C
Theorems
Vertical Angles
Theorem
If two angles are vertical, then they are congruent.
Complement
Theorem
If two angles form a right angle,
then they are complementary.
Right Angle → Complementary
Supplement
Theorem
If two angles form a linear pair,
then they are supplementary.
Linear pair → Supplementary
Congruent
Complements
Theorem
If ∠A is complementary to ∠B and ∠C
is complementary to ∠B, then ∠A ≅ ∠C
Congruent
Supplements
Theorem
If ∠A is supplementary to ∠B and ∠C
is supplementary to ∠B, then ∠A ≅ ∠C
© Gina Wilson (All Things Algebra), 2014
Practice!
Justify each of the following statements using a definition,
definition, theorem
theorem or postulate.
postulate.
1. If ∠A is a right angle, then m∠A = 90°
_________________________________________
2. If ∠X is supplementary to ∠Y and ∠X
is supplementary to ∠Z , then ∠X ≅ ∠Z.
_________________________________________
3. If
_________________________________________
1
then, ∠1 ≅ ∠2
2
4. If m∠P + m∠Q = 90°,
then ∠P and ∠Q are complementary.
_________________________________________
5. If ∠M and ∠N form a right angle, then
then ∠M and ∠N are complementary.
_________________________________________
m
6. Given:
1
l
If l ⊥ m, then
∠1 is a right angle.
_________________________________________
7. If ∠W and ∠X are supplementary,
then m∠W + m∠X = 180°.
_________________________________________
8. If ∠L is complementary to ∠M and ∠N
is complementary to ∠M , then ∠L ≅ ∠N.
_________________________________________
9. If ∠A and ∠B form a linear pair, then
then ∠A and ∠B are supplementary.
_________________________________________
10. If ∠N and ∠P are complementary, then
m∠N + m∠P = 90°.
11.
K
Given:
_________________________________________
_________________________________________
L
J
M
m∠JKM + m∠MKL = m∠JKL
12.
If m∠R = m∠S, then ∠R ≅ ∠T
_________________________________________
© Gina Wilson (All Things Algebra), 2014
ANGLE PROOFS Guide!
Guide!
Directions: Use the reasons below to complete proofs 1-6.
1
Supplement Theorem
Substitution
Given
Definition of Supplementary
Angles
• Definition of Congruence
• Definition of Supplementary
Angles
• Given
•
•
•
•
• Definition of a Right Angle
• Definition of
Complementary Angles
• Given
• Transitive Property
4
• Given
• Congruent Complements
Theorem
• Complement Theorem
• Given
• Definition of Complementary
Angles
•
•
•
•
•
•
•
Given
Substitution
Definition of Congruence
Transitive Property
Definition of Congruence
2
•
•
•
•
•
Definition of Angle Bisector
Given
Given
Transitive Property
Definition of Angle Bisector
•
•
•
•
Definition of Congruence
Transitive Property
Subtraction Property
Definition of Complementary
Angles
Definition of Congruence
Substitution
Definition of Complementary
Angles
Given
Vertical Angles Theorem
Given
•
•
•
•
•
•
© Gina Wilson (All Things Algebra), 2014
Supplement Theorem
Substitution
Given
Definition of Supplementary
Angles
• Definition of Congruence
• Definition of Supplementary
Angles
• Given
•
•
•
•
• Definition of a Right Angle
• Definition of
Complementary Angles
• Given
• Transitive Property
4
3
6
5
Directions: Use the reasons below to complete proofs 1-6.
1
2
3
ANGLE PROOFS Guide!
Guide!
• Given
• Congruent Complements
Theorem
• Complement Theorem
• Given
• Definition of Complementary
Angles
•
•
•
•
•
Definition of Angle Bisector
Given
Given
Transitive Property
Definition of Angle Bisector
•
•
•
•
Definition of Congruence
Transitive Property
Subtraction Property
Definition of Complementary
Angles
Definition of Congruence
Substitution
Definition of Complementary
Angles
Given
Vertical Angles Theorem
Given
6
5
•
•
•
•
•
•
•
Given
Substitution
Definition of Congruence
Transitive Property
Definition of Congruence
•
•
•
•
•
•
© Gina Wilson (All Things Algebra), 2014
ANGLE PROOFS
Directions:
Complete the proofs below by giving the missing statements and reasons.
P
❶ Given:
∠PQR is a right angle
Prove: ∠PQS and ∠SQR are complementary
S
Q
Statements
R
Reasons
1. ∠PQR is a right angle
1.
2. m∠PQR = 90°
2.
3. m∠PQS + m∠SQR = m∠PQR
3.
4. m∠PQS + m∠SQR = 90°
4.
5. ∠PQS and ∠SQR are complementary
5.
❷ Given:
∠2 ≅ ∠3; ∠1 and ∠2 form a linear pair
Prove: ∠1 and ∠3 are supplementary
Statements
3
2
Reasons
1. ∠2 ≅ ∠3
1.
2. m∠2 = m∠3
2.
3. ∠1 and ∠2 form a linear pair
3.
4. ∠1 and ∠2 are supplementary
4.
m∠1 + m∠2 = 180°
5.
6. m∠1 + m∠3 = 180°
6.
7. ∠1 and ∠3 are supplementary
7.
5.
1
❸ Given:
∠1 and ∠2 form a right angle; m∠1 + m∠3 = 90°
Prove: ∠2 ≅ ∠3
Statements
1
2
3
Reasons
1. ∠1 and ∠2 form a right angle
1.
2. ∠1 and ∠2 are complementary
2.
3. m∠1 + m∠3 = 90°
3.
4. ∠1 and ∠3 are complementary
4.
5. ∠2 ≅ ∠3
5.
© Gina Wilson (All Things Algebra), 2014
E
❹ Given:
BE bisects ∠ABD; BD bisects ∠EBC
Prove: ∠ABE ≅ ∠DBC
D
A
C
B
Statements
Reasons
1. BE bisects ∠ABD
1.
2. ∠ABE ≅ ∠EBD
2.
3.
BD bisects ∠EBC
3.
4. ∠EBD ≅ ∠DBC
4.
5. ∠ABE ≅ ∠DBC
5.
❺ Given:
U
∠RSU ≅ ∠VST
Prove: ∠RSV ≅ ∠UST
R
V
S
Statements
Reasons
1. ∠RSU ≅ ∠VST
1.
2. m∠RSU = m∠VST
2.
3. m∠RSU + m∠USV = m∠RSV
3.
4. m∠VST + m∠USV = m∠UST
4.
5. m∠RSU + m∠USV = m∠UST
5.
6. m∠RSV = m∠UST
6.
7. ∠RSV ≅ ∠UST
7.
❻ Given:
4
∠1 and ∠2 are complementary
∠3 and ∠4 are complementary
2
Statements
Reasons
1. ∠1 and ∠2 are complementary
1.
2. ∠3 and ∠4 are complementary
2.
3. m∠1 + m∠2 = 90°
3.
4. m∠3 + m∠4 = 90°
4.
5. ∠2 ≅ ∠3
5.
m∠2 = m∠3
3
1
Prove: ∠1 ≅ ∠4
6.
T
6.
7. m∠1 + m∠2 = m∠3 + m∠4
7.
8. m∠1 + m∠3 = m∠3 + m∠4
8.
9. m∠1 = m∠4
9.
10. ∠1 ≅ ∠4
10.
© Gina Wilson (All Things Algebra), 2014
❼ Given:
∠1 ≅ ∠4; ∠4 and ∠5 form a linear pair
Prove: ∠1 and ∠5 are supplementary
1
5
Statements
2
3
4
Reasons
1. ∠1 ≅ ∠4
1.
2.
2. Definition of Congruence
3.
3. Given
4. ∠4 and ∠5 are supplementary
4.
5.
5. Definition of Supplementary Angles
6.
6. Substitution
7. ∠1 and ∠5 are supplementary
7.
❽ Given:
∠1 and ∠2 form a linear pair; m∠2 + m∠3 = 180°
Prove: ∠1 ≅ ∠3
Statements
1
3
2
Reasons
1. ∠1 and ∠2 form a linear pair
1.
2.
2. The Supplement Theorem
3.
3. Given
4.
4. Definition of Supplementary Angles
5.
∠1 ≅ ∠3
5.
B
❾ Given:
AB ⊥ BC ; ∠2 and ∠3 are complementary
Prove: ∠1 ≅ ∠2
1 3
2
A
Statements
D
C
Reasons
1. AB ⊥ BC
1.
2. ∠ABC is a right angle
2.
3.
3. Definition of a Right Angle
4. m∠1 + m∠3 = m∠ABC
4.
5. m∠1 + m∠3 = 90°
5.
6.
6. Definition of Complementary Angles
7.
7. Given
8. ∠1 ≅ ∠2
8.
© Gina Wilson (All Things Algebra), 2014
Name: ___________________________________
Unit 2: Logic & Proof
Date: ________________________ Bell: ______
Homework 8: Angle Proofs
** This is a 2-page document! **
Given each definition or theorem, complete each statement.
1. Definition of Congruence:
If ∠D ≅ ∠E, then ___________________________________
2. Definition of Complementary Angles:
If m∠1 + m∠2 = 90°, then _____________________________________________
3. Definition of Supplementary Angles:
If ∠P and ∠Q are supplementary angles, then ___________________________________
4. Definition of a Right Angle:
If m∠ JKL = 90°, then ___________________________________
5. Vertical Angles Theorem:
If ∠3 and ∠4 are vertical angles, then __________________________
6. Complement Theorem:
If ∠S and ∠T form a right angle, then _____________________________________________
7. Supplement Theorem:
If ∠X and ∠Y form a linear pair, then _____________________________________________
8. Congruent Complements Theorem: If ∠1 is complementary to ∠2
and ∠2 is complementary to ∠4, then ___________________
9. Congruent Complements Theorem: If ∠J is supplementary to ∠K
and ∠J is supplementary to ∠L, then ___________________
Complete the proofs below by filling in the missing statements and reasons.
10. Given: ∠1 and ∠2 form a linear pair;
∠1 and ∠3 are supplementary
Prove: ∠2 ≅ ∠3
1
Statements
Reasons
1. ∠1 and ∠2 form a linear pair
1.
2. ∠1 and ∠2 are supplementary
2.
3.
m∠1 + m∠2 = 180°
4. ∠1 and ∠3 are supplementary
3
2
3.
4.
5.
m∠1 + m∠3 = 180°
5.
6.
m∠1 + m∠2 = m∠1 + m∠3
6.
7. m∠2 = m∠3
7.
8. ∠2 ≅ ∠3
8.
© Gina Wilson (All Things Algebra), 2014
11. Given:
M
KM bisects ∠JKL
J
1
2
Statements
L
Prove: m∠MKL = m∠JKL
K
Reasons
1. KM bisects ∠JKL
1.
2. m∠JKM = m∠MKL
2.
3. m∠JKM + m∠MKL = m∠JKL
3.
4. m∠MKL + m∠MKL = m∠JKL
4.
5. 2m∠MKL = m∠JKL
5.
6. m∠MKL =
12. Given:
1
m∠JKL
2
6.
D
BD ⊥ BC ; ∠ABD ≅ ∠DBE
E
Prove: ∠ABD and ∠EBC are complementary
Statements
A
B
C
Reasons
1. BD ⊥ BC
1.
2. ∠DBC is a right angle
2.
3. m∠DBC = 90°
3.
4. m∠DBE + m∠EBC = m∠DBC
4.
5. m∠DBE + m∠EBC = 90°
5.
6. ∠ABD ≅ ∠DBE
6.
7. m∠ABD = m∠DBE
7.
8. m∠ABD + m∠EBC = 90°
8.
9. ∠ABD and ∠EBC are complementary
9.
13. Given: ∠1 and ∠4 form a linear pair;
∠1 and ∠2 are supplementary
1
4
3
2
Prove: ∠3 ≅ ∠4
Statements
Reasons
1. ∠1 and ∠4 form a linear pair
1.
2.
2. Supplement Theorem
3.
3. Given
4.
4. Congruent Supplements Theorem
5. ∠2 ≅ ∠3
5.
6.
6. Transitive Property
© Gina Wilson (All Things Algebra), 2014
Name: ________________________________________
Geometry
Date: ______________________________ Per: ______
Unit 2: Logic & Proof
Quiz 2-3: Segment & Angle Proofs
Write the letter of the property, definition, or postulate that justifies each statement.
_____1. QR = QR
A.
B.
C.
D.
E.
F.
G.
H.
I.
J.
K.
_____2. If AB = CD, then AB + EF = CD + EF
_____3. If RS + TU = XY and TU = WV, then RS + WV = XY
_____4. JK = LM, then JK ≅ LM
_____5. If AB ≅ BC and BC ≅ CE , then AB ≅ CE
_____6. If 2XY = YZ, then XY = ½YZ
_____7. If Q is between P and R, then PQ + QR = PR
_____8. If 2KL = KL + MN, then KL = MN
Subtraction Property of Equality
Multiplication Property of Equality
Division Property of Equality
Substitution Property
Reflexive Property (of = or ≅)
Symmetric Property (of = or ≅)
Transitive Property (of = or ≅)
Definition of Congruence
Definition of Midpoint
Complete the proofs below by filling in the missing statements and reasons.
9.
Given: B is the midpoint of AC , AB ≅ CD
A
B
C
D
Prove: C is the midpoint of BD
Statements
Reasons
1. B is the midpoint of AC
1.
2. AB = BC
2.
3. AB ≅ CD
3.
4. AB = CD
4.
5. BC = CD
5.
6. C is the midpoint of BD
6.
R
10. Given: PS = RT, PQ = ST
Prove: QS = RS
Q
P
S
T
Statements
Reasons
1.
PS = RT, PQ = ST
1.
2.
PQ + QS = PS
2.
3.
ST + QS = RT
3.
4.
RS + ST = RT
4.
5.
ST + QS = RS + ST
5.
6. QS = RS
6.
© Gina Wilson (All Things Algebra), 2014
Write the letter of the definition, theorem, or postulate that justifies each statement.
_____11. If m∠ ABC = 90°, then ∠ABC is a right angle.
_____12. If m∠3 + m∠4 = 180°, then ∠3 and ∠4 are
supplementary angles.
_____13. If m∠PQR = m∠RST, then ∠PQR ≅ ∠RST
_____14. If ∠1 and ∠2 form a right angle, then ∠1 and ∠2
are complementary angles.
_____15. If ∠X and ∠Y are supplementary and ∠X and ∠Z
are supplementary, then ∠Y ≅ ∠Z
A.
B.
C.
D.
E.
F.
G.
H.
I.
J.
K.
L.
Definition of Congruence
Definition of Angle Bisector
Definition of Complementary ∠’s
Definition of Supplementary ∠’s
Definition of Perpendicular
Definition of a Right Angle
Vertical Angles Theorem
Complement Theorem
Supplement Theorem
Congruent Complements Theorem
Congruent Supplements Theorem
_____16. If ∠J and ∠K are vertical angles, then ∠J ≅ ∠K
_____17. If ∠ABC and ∠DEF are complementary angles, then m∠PQR + m∠RST = 90°.
_____18. If ∠5 and ∠6 form a linear, then ∠5 and ∠6 are supplementary angles.
Complete the proofs below by filling in the missing statements and reasons.
19. Given: ∠1 and ∠2 form a linear pair; ∠1 ≅ ∠3
Prove: ∠2 and ∠3 are supplementary
Statements
Reasons
1. ∠1 and ∠2 form a linear pair
1.
2. ∠1 and ∠2 are supplementary
2.
3. m∠1 + m∠2 = 180°
3.
4. ∠1 ≅ ∠3
4.
5. m∠1 = m∠3
5.
6. m∠3 + m∠2 = 180°
6.
7. ∠2 and ∠3 are supplementary
7.
20. Given: ∠PQR is a right angle;
∠TQP and ∠SQR are complementary angles
Prove: ∠PQS ≅ ∠TQP
1. ∠PQR is a right angle
m∠PQR = 90°
P
S
T
Q
Statements
2.
2
1
3
R
Reasons
1.
2.
3. m∠PQS + m∠SQR = ∠PQR
3.
3. m∠PQS + m∠SQR = 90°
4.
3. ∠PQS and ∠SQR are complementary angles
5.
4. ∠TQP and ∠SQR are complementary angles
6.
7. ∠PQS ≅ ∠TQP
7.
© Gina Wilson (All Things Algebra), 2014
Unit 2 Test Study Guide
(Logic & Proof)
Name: _________________________________________
Date: _________________________ Block: __________
Topic #1: Conjectures & Counterexamples
Directions: Determine if the conjecture is true or false. If false, provide a counterexample.
1. The product of any two prime numbers is always odd.
2. If two angles are complementary, then both angles must be acute.
3. The square of a number is always larger than the number.
4. Two lines always intersect at a point.
Topic 2: Compound Statements & Truth Tables
Conjunction:
True when: ___________ statements are ___________.
Disjunction:
True when: _____ ____________ ________ statement is ___________.
Use the following statements to write compound statements. Determine the truth value.
p: All vegetables are green.
q: Vertical angles are congruent.
r: All integers are natural numbers.
5. p ∨ q: _________________________________________________________________________________________
___________________________________________________________________ Truth Value: _________
6. q ∧ r: _________________________________________________________________________________________
___________________________________________________________________ Truth Value: _________
7. p ∨ ~q: ________________________________________________________________________________________
___________________________________________________________________ Truth Value: _________
8. q ∧ ~r: ________________________________________________________________________________________
___________________________________________________________________ Truth Value: _________
© Gina Wilson (All Things Algebra), 2014
Directions: Complete each truth table.
9. ~p ∨ ~q
10. ~r ∨ (~p ∧ q)
Topic 3: Conditional Statements
Conditional:
Inverse:
Converse:
Contrapositive:
Use the following statements to write conditional statements. Determine the truth value.
p: two angles form a linear pair; q: they are adjacent angle
11. Conditional: _______________________________________________________________________
_____________________________________________________Truth Value: _______
12. Inverse: _________________________________________________________________________
_______________________________________________________Truth Value: ________
13. Converse: _________________________________________________________________________
______________________________________________________Truth Value: ________
14. Contrapositive: _____________________________________________________________________
___________________________________________________Truth Value: ________
© Gina Wilson (All Things Algebra), 2014
Topic 4: Bi-Conditional Statements
Bi-Conditional:
True when both conditional (p → q) and converse (q → p) are true!
Use the given statements to write the conditional, converse, and bi-conditional. Determine
the truth value of the bi-conditional. Explain.
15. p: a quadrilateral is a square; q: it has four congruent sides
Conditional: __________________________________________________________________________
Converse: ___________________________________________________________________________
Bi-Conditional: ________________________________________________________________________
Truth Value: __________________________________________________________________________
Topic 5: Deductive Reasoning
Law of Detachment:
Law of Syllogism:
Use the Law of Detachment or Law of Syllogism to make a conclusion. If a conclusion
cannot be made, write “no valid conclusion”.
16. Given: If it is Thursday, then Emma has drama club.
If Emma has drama club, then she must take the late bus.
Conclusion: _________________________________________________________________________
17. Given: If two angles are vertical, then they are congruent.
Two angles are congruent.
Conclusion: _________________________________________________________________________
18. Given: If the football team raises at least \$5000, then they can buy new uniforms.
The football team raised \$5800.
Conclusion: _________________________________________________________________________
19. Given: If you return a book to the library late, then you will pay a fee.
If you do not pay your library fees, then you may lose your library card.
Conclusion: ______________________________________________________________________
© Gina Wilson (All Things Algebra), 2014
Topic 6: Venn Diagrams
Draw a Venn Diagram to represent each relationship.
20. All linear pairs are adjacent.
21. No rectangles are trapezoids. 23. Some multiples of 3 are also
multiples of 4.
24. The Venn Diagram below shows the
number of boy scouts in a certain troop
who have earned the following merit
badges: camping, swimming, and first aid.
Swimming
Camping
4
10
6
2
7
3
5
2
First Aid
_______a. How many boy scouts have a first aid
_______b. How many boy scouts have a camping
_______c. How many boy scouts have a camping
_______d. How many boy scouts have a swimming
_______e. How many boy scouts do not have a
_______f. How many boy scouts are in the troop?
Topic 7: Algebraic Proof
KNOW THE FOLLOWING:
Distributive, Substitution, Reflexive, Symmetric, and Transitive
25. Given: 8(x – 1) = 5x – 35;
Prove: x = -9
Statements
Reasons
© Gina Wilson (All Things Algebra), 2014
Topic 8: Segment Proofs
KNOW THE FOLLOWING:
Segment Addition Postulate, Definition of Midpoint, Definition of Congruence,
Properties of Congruence (Reflexive, Symmetric, and Transitive)
26. Given: Q is the midpoint of PR
R is the midpoint of QS
Prove: PQ =
1
2
P
S
Reasons
1. Q is the midpoint of PR
1.
2. R is the midpoint of QS
2.
3. PQ = QR; QR = RS
3.
4. PQ = RS
4.
5. PQ + PQ = PQ + RS
5.
6. 2PQ = PQ + RS
6.
7. 2PQ = QR + RS
7.
8. QR + RS = QS
8.
9. 2PQ = QS
9.
1
2
R
QS
Statements
10. PQ =
Q
QS
10.
A
27. Given: AB ≅ CD ; CE ≅ AE
D
Prove: ED ≅ EB
E
B
C
Statements
Reasons
1. AB ≅ CD ; CE ≅ AE
1.
2. AB = CD; CE = AE
2.
3. AE + EB = AB; CE + ED = CD
3.
4. CE + EB = CD
4.
5. CE + ED = CE + EB
5.
6. ED = EB
6.
7. ED ≅ EB
7.
© Gina Wilson (All Things Algebra), 2014
Topic 9: Angle Proofs
8
KNOW THE FOLLOWING:
Definition of Congruence
Definition of Angle Bisector
Definition of Complementary Angles
Definition of Supplementary Angles
Defining of Perpendicular
Definition of Right Angle
Vertical Angles Theorem
Complement Theorem
Supplement Theorem
Congruent Complements Theorem
Congruent Supplements Theorem
28. Given: ∠1 and ∠2 are complementary; ∠1 ≅ ∠4
Prove: ∠3 and ∠4 are complementary
4
3
2
1
Statements
Reasons
1. ∠1 and ∠2 are complementary
1.
2. m∠1 + m∠2 = 90°
2.
3. ∠1 ≅ ∠4
3.
4. m∠1 = m∠4
4.
5. ∠2 ≅ ∠3
5.
6. m∠2 = m∠3
6.
7. m∠4 + m∠3 = 90°
7.
8. ∠3 and ∠4 are complementary
8.
29. Given: ∠JKM and ∠MKL form a linear pair;
∠JKM and ∠LKN are supplementary
Prove: ∠MKL ≅ ∠LKN
M
J
K
L
N
Statements
Reasons
1. ∠JKM and ∠MKL form a linear pair
1.
2. ∠JKM and ∠MKL are supplementary
2.
3. ∠JKM and ∠LKN are supplementary
3.
4. ∠MKL ≅ ∠LKN
4.
© Gina Wilson (All Things Algebra), 2014
Name: _________________________________________
Unit 2 Test
Logic & Proof
Date: _______________________________ Bell: ______
For questions 1-2, determine if the conjectures are true or false.
If false, provide a counterexample.
1. All perfect squares are divisible by 2. ____________________________________________
2. Multiples of 3 are always multiples of 6. __________________________________________
3. Which diagram provides a counterexample to the statement below?
“Supplementary angles are never congruent.”
A.
B.
C.
1
1
2
2
D.
1
2
1
2
Use the statements below to answer questions 4-5.
p: Memorial Day is in July; q: Parallel lines never intersect.
5. Check the statements below that are false.
4. Which represents the symbolic notation of
the compound statement below?
Memorial Day is in July or parallel lines
“Memorial Day is in July or
parallel lines intersect.”
A. p ∧ q
B. p ∧ ~q
C. p ∨ q
D. p ∨ ~q
intersect.
Memorial Day is not in July and parallel lines
never intersect.
Memorial Day is not in July and parallel lines
intersect.
Memorial Day is in July or parallel lines
never intersect.
Memorial Day is not in July or parallel lines
intersect.
6. Complete a truth table given the compound statement below.
Given: (p ∨ ~r) ∧ ~q
© Gina Wilson (All Things Algebra), 2014
Use the statements below to complete questions 7-10.
p: a month begins with the letter M; q: it has 31 days
7. Conditional: ______________________________________________________________________
____________________________________________________________Truth Value: __________
8. Inverse: _________________________________________________________________________
____________________________________________________________Truth Value: ___________
9. Converse: _______________________________________________________________________
____________________________________________________________Truth Value: __________
10. Contrapositive: ___________________________________________________________________
____________________________________________________________Truth Value: __________
11. Which symbolic notation represents the
inverse of a conditional statement?
A. q → p
B. p → q
C. ~q → ~p
D. ~p → ~q
12. Which bi-conditional statement below is true?
A. Angles are congruent if and
only if they are vertical angles.
B. Angles are supplementary if and
only if their sum is 180°.
C. A number is a whole number if
and only if it is a natural number.
D. Points are collinear if and
only if they are coplanar.
State whether the Law of Detachment or the Law of Syllogism was used to draw the
conclusion from the given statements. If it’s an invalid conclusion, write invalid.
13. Given: If you live in Orlando, then you live in Florida.
Given: Morgan does not live in Orlando.
Conclusion: Morgan does not live in Florida.
14. Given: If two angles in a triangle are 40° and 30°, then the third angle measures 110°.
Given: If a triangle has an angle that measures 110°, then it is an obtuse triangle.
Conclusion: If two angles in a triangle are 40° and 30°, then it is an obtuse triangle.
© Gina Wilson (All Things Algebra), 2014
15. Given: If you go fishing, then you will need to bring your fishing rod.
Given: If you go fishing, then you will need to bring bait.
Conclusion: If you bring your fishing rod, then you will need to bring bait.
16. Given: If you are at least 25 years old, then you can rent a car.
Given: Anna is 28 years old.
Conclusion: Anna can rent a car.
17. Which statement best represents the Venn diagram below?
A. Some politicians are democrats.
Politicians
B. All politicians are democrats.
C. All democrats are politicians.
D. Some democrats are politicians.
Democrats
18. The Venn diagram below shows the relationship between college students who take at least 15
credits and work a part-time job. Which diagram shows students who are take at least 15
credits and do not work a part-time job?
A.
B.
15 Credits
Part-Time
Job
C.
15 Credits
Part-Time
Job
The Venn diagram below shows the number
of students in Mrs. Crane’s homeroom who
were on Honor Roll, taking advanced classes
or had perfect attendance for the first quarter.
Classes
Honor Roll
2
9
3
4
1
5
15 Credits
D.
Part-Time
Job
15 Credits
Part-Time
Job
19. How many students are taking advanced
classes?
20. How many students are on Honor Roll or had
perfect attendance?
2
4
Perfect Attendance
21. How many students are taking advanced classes
and on Honor Roll, but did not have perfect
attendance?
© Gina Wilson (All Things Algebra), 2014
22. Given: -3(2x + 7) = -29 – 4x;
Prove: x = 4
Statements
Reasons
23. Which reason justifies the statement below?
24. Which reason justifies the statement below?
“If PQ +
RS = PS and RS = XY,
then PQ + XY = PS”
A.
B.
C.
D.
“If CD = EF, then
Symmetric Property
Substitution Property
Transitive Property
A.
B.
C.
D.
25. Which reason justifies the statement below?
Definition of Congruence
Reflexive Property
Symmetric Property
Definition of Midpoint
26. Which reason justifies the statement below?
“If AB +
BC = AC and AC = EF + FG,
then AB + BC = EF + FG ”
A.
B.
C.
D.
“If
Subtraction Property
Transitive Property
CD ≅ EF ”
A.
B.
C.
D.
JK + LM = NP + LM,
then JK = NP ”
Reflexive Property
Substitution Property
Subtraction Property
27. Given: WY ≅ XZ
W
Prove: WX ≅ YZ
Statements
WY ≅ XZ
1.
2.
WY = XZ
2.
XZ = WZ; WY + YZ = WZ
Y
Z
Reasons
1.
3. WX +
X
3.
4.
XZ + YZ = WZ
4.
5.
WX + XZ = XZ + YZ
5.
6.
WX = YZ
6.
7.
WX ≅ YZ
7.
© Gina Wilson (All Things Algebra), 2014
28. Given: D is the midpoint of CE
Prove: DE = ½CE
C
Statements
D is the midpoint of CE
1.
2.
CD = DE
2.
3.
CD + DE = DE + DE
3.
4.
CD + DE = 2DE
4.
5.
CD + DE = CE
5.
7.
= CE
6.
DE = ½CE
7.
29. Which reason justifies the statement below?
A
Given:
D
B
A.
B.
C.
D.
C
30. Which reason justifies the statement below?
“m∠ABD + m∠DBC
= m∠ABC”
“If
Definition of Angle Bisector
Definition of Congruence
A.
B.
C.
D.
31. Which reason justifies the statement below?
1
then, ∠1 ≅ ∠2”
2
Definition of Congruence
Supplement Theorem
Definition of Vertical Angles
32. Which reason justifies the statement below?
m
“If ∠A and ∠B are complementary,
then m∠
∠A + m∠
∠B = 90°”
A.
B.
C.
D.
E
Reasons
1.
6. 2DE
D
Given:
The Complement Theorem
Definition of a Right Angle
Congruent Complements Theorem
Definition of Complementary Angles
A.
B.
C.
D.
1
“ If l ⊥ m, then
∠1 is a right angle”
l
Definition of Complementary Angles
Definition of a Right Angle
Definition of Perpendicular
The Complement Theorem
33. Given: ∠1 and ∠3 are form a right angle
∠1 and ∠2 are complementary
Prove: ∠2 ≅ ∠3
B
1 3
A
Statements
2
D
C
Reasons
1. ∠1 and ∠3 are form a right angle
1.
2. ∠1 and ∠3 are complementary
2.
3. ∠1 and ∠2 are complementary
3.
4. 2 ≅ ∠3
4.
© Gina Wilson (All Things Algebra), 2014
34. Given: KM bisects ∠JKN; KN bisects ∠MKL
Prove: ∠JKM ≅ ∠NKL
M
N
J
L
K
Statements
Reasons
1. KM bisects ∠JKN
1.
2. ∠JKM ≅ ∠MKN
2.
3. KN bisects ∠MKL
3.
4. ∠MKN ≅ ∠NKL
4.
5. ∠JKM ≅ ∠NKL
5.
35. Given: ∠1 and ∠2 form a linear pair; ∠2 ≅ ∠4
Prove: ∠1 and ∠3 are supplementary
Statements
1.
2. ∠1 and ∠2 are supplementary
2.
m∠1 + m∠2 = 180°
4
3.
4. ∠2 ≅ ∠4
4.
5. ∠3 ≅ ∠4
5.
6. ∠2 ≅ ∠3
6.
7.
m∠2 = m∠3
7.
8.
m∠1 + m∠3 = 180°
8.
9. ∠1 and ∠3 are supplementary
3
2
Reasons
1. ∠1 and ∠2 form a linear pair
3.
1
9.
© Gina Wilson (All Things Algebra), 2014
Unit 2: Logic & Proof
Name:
Homework 2: Compound Statements
Bell:
Date:
** This is a 2-page document! **
Directions: Use the statements below along with the diagram to write compound statements.
Then find its truth value.
F
(h) p: Points C, E, and B are collinear.
A
q: LAECZDEB
E
r: EF is the angle bisector of ZAED
s: ZBEC is an acute angle.
D
B
1.pvq:
i. and i
!YtS
Th
L L
c1L+e
pA
g
Truth Value:
I el .
r 211s L )
Truth Value:
0'C1
13 prt
r1I11m)
FF
4. rV
r + 0 r', 0( i je nnale .
(id
qAr:
6. pV
Truth Value:
r
Truth Value:
I
q:
P0ThtS
Truth Value:
C.,
€, Ot)d
t!
i.
7. 'rV
1
s:
I
qAs:
9. pV
(/C1
fi
!,
'ti
b
S.
F
s:
1
5.
/
E Ci
n1
'
!ivGr
13.
2. q A s:
3.
ir
s
C_
hp
t
Truth Value:
I
*he
e
0 (O
C"(i <E(.
!
r
TruthValue:
J i30 ,2 (i/iU
Of
A
Truth Value:
F
Truth Value:
/
o
.
Directions: Complete each truth table.
10.pAr
11. q V s
c1\i S
c
r
T
_
F
F
F
F
T
F
P
F
_
13. r"pAs
12. riqVr
C7
T
_
Vr
C7
F
R
F
F
T
FT
•
P
E
14.
's:
1
T
E
F
1
F
F
•i F
E
T
F
F
F
7-
rA (pVq)
p
T
F
T
F
F
r
T
Vc
rACVc
F
I
F
F
Ti
F F
F
F
F
F
F
F
F
FT
T
F
r
P
T
F
F
FE
15.
F
r
1-
Fr
(pv"r)A"q
r
T-
r
T-
F
FF
F
Fr
FrF
F
F
F
1
F
FF
1FF1T
Name:
Class:
opic:
Date:
Main Ideas/Questions
Notes
-___ form.
I
Cj
Symbolic Form:
Conditional
Statement
\4
A statement that can be written in
•
or, "p
q".
•
The tillXtY3S
UT
is the phrase immediately following the word
•
The CDI'Cft)
is the phrase immediately following the word
uC'
Thr
Identify the hypothesis and conclusion of the following conditional statements:
Examples
1. If you live in Nashville, then you live in Tennessee.
UVe
Hypothesis:
''k
Conclusion:
"4(A ttyf
JI\
iV)SjiiIi
,v
2. If the sum of the measures of two angles is 90 °, then they are
complementary angles.
Hypothesis:' -
he
Conclusion:
ih.L.j CEVe
'-;-
if -rh Ov)l?c
-f
('IY))\/7 o flhYtth(1'
U41 le,c.
3. If a quadrilateral is a square, then it has four right angles.
Hypothesis:
Conclusion:
a sgure
f'r r± mgie.
ft hO.
Writing Conditional Statements: Write the following statements in if-then form.
4. An obtuse angle has a measure greater than 900.
ir
on ctng1
c
097..2c
-I/- er
0 ha
tyw
e giea/ei-
j/n
(Y.
5. All numbers divisible by 4 are also divisible by 2.
rmbfr
'Ji'n ri
j
i
is D chViL)
1y
6. States on the east coast border the Atlantic Ocean.
if a ct(H-/ if
v-
he
(rYis
±1•i ,4 /'k7IrS -th i102
7. Valentine's Day is in February.
L
8. Prime numbers only have two factors, 1 and itself.
1±
hI
3I
YCQt
/
(nd
0cv1
Con&widw
r 3 (t)flLJ
Formed by
Inverse
Symbolic form:
Formed by
C
\J
p
Y-C
Symbolic form:
the hypothesis and conclusion.
C
•
p
Formed by ('\egOl\ r\q
hypothesis and donclusioh.
Contrapositive
the hypothesis and conclusion.
and
CQ
the
Symbolic form:
Directions: Write the inverse, converse, and contrapositive of the following conditional statements.
Determine the truth value. If false, provide a counterexample.
1. If it is Saturday, then there is no school.
•
•
if Ir- ) ñDt
Truth Value: j- )J ',
Converse:
Truth Value:
•
+hr\ 'f her
Inverse:
if ThItY'
,s
ie
IS ('\D
hoD) ) thI) ii
L.S
tctI__
±hU
i
Contrapositive:
L
( hc'l
"Iell il
12t
I
Truth Value:
2. If the product of two numbers is odd, then both numbers must be odd.
•
cf'
11 '±F
Inverse:
both rUmJ,Tc
•
Converse:
tJD .yil h ic s • v r, li-)(I rl
CnL- ),S-r k,VeT'. TruthValue: Fu/. EX : '-L7
Lf ±'AJ3
Odd
•
Contrapositive:
(kT
Truth Value:
I ( 1VJ D tTh)YY' I'
'1L'm - ,ndur1
/(i.
(1
Ti'
evr' 1 I nC
ihIr p'iYiiYi
Truth Value:
is 25°F, then it is below freezing.
erperU1Jie Ic
5" F l'ñ )i ii'
• Inverse: i'I ±K
Truth Value: h')ISe. H (uid
2.1h3
r -- be tOv)
3. If the temperature
•
Converse:
_rt IS ke low 'E?'i 29) fTr\ -) h
S
•
Contrapositive:
Truth Value:
142 ± l
T°
i'flprQiXP
tQIS. !t (uid kLi
0 t VP- /0 ti * e e i..i - ij ther
r1j(
k(-e c 1'01 cQ5- Truth Value:
Bi-conDitional statements
t&\nW
i1e c mj
Srv*'otic or
Yli3'\
üf a)
C/
(p -* q) A (q —*p):
UiYi it
Bi-Conditional statements are true when
CO'Yiftora)
cvrcc
and
are
IcUE.
Examples: Given the bi-conditional statement below, write both the conditional and converse.
Determine the truth value of the bi-conditional. Explain why or why not.
1. Two angles are supplementary if and only if the sum of their measures is 1800.
Conditional
Converse:
f
5i)Y
r
Thi'
, *)r\ -
T(L)e b) -
Truth Value?
WL)
Thu orecpp)r Jft'j. (7)
ate
C()rY1r Or12) r r'c) COrve r
2. I wear my snow boots if and only if it snows.
±ii1\ i
Conditional:
Converse:
Truth Value?
II
I
flA)
Uiñr rri1 <r\
ft) /
3. x2 = 25 if and only ifx = 5.
Conditional:
Converse:
Truth Value?
+cice.
fbk h)
(x cdcô/ -
4. I will get 10% off if and only if I spent at least \$75.
Conditional: i --
P
I (')1 lcio oPf
Then
.
C7
Converse:
Truth Value?
Y)
Tht
two
hTh (YJIY/Th (fYi!
(7i
Cl t?
.
{A grapblc organizer for logic statements)
Compound Statements
Directions: Use the statements below to write the compound statements below.
(1) p: Vertical angles are congruent.
u)
q: 15 is a prime number.
\1
co\$xicflon
A
[
VC
(
Cl
(
]
I' (
c\
ccnovufind
J
Truth value:
V exi cc I
D4nc11On
[
ii
('
I cCtre
Truth value:
\
]
Condlilond Statements
Directions: Use the statements below to write the conditional statements below.
p: If it is St. Patrick's Day.
q: It is March.
Condlhlond
I
s
mn
Inverse
[
1
4 ;
I
[c1
Conlrcposttive
-V
eDTn C
If
pj
BI-Condlilond
c
.
Truth value:
rrr- St Piy/i c f)o9 i-hen
Converse
~
C_4
(
]
L
L
Pd-'iKs
.
flr(j
S
)i iS
g
t.
FO/.
-
+. pmVKS,
If I
1'iP\
Truth value:
'Truth value:
I
--
]
Truth value:
-S (Torch
.
Truth value:
mdOn/LJ fr 1
CIIJt
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