Geometry: Logic
Day
Topics
1
Negations, conjunctions and disjunctions
HW Logic - 1
2
Conditionals
HW Logic - 2
3
Logical reasoning
HW Logic - 3
4
Related conditionals
5
Logical reasoning (again)
6
Biconditionals, definitions ***QUIZ*** HW Logic - 6
7
Logic proofs
HW Logic - 7
8
Practice
HW Logic - 8
9
Review Day ***QUIZ***
HW Logic - Review
10
***TEST***
***QUIZ***
Homework
HW Quiz
Grade Grade
HW Logic - 4
HW Logic - 5
Record your homework, quiz and test grades (including make-up quizzes and test
corrections) and save this page. Please do not ask “What are my grades?”
SAVE your homeworks after you get them back.
Geometry HW: Logic - 1
1. Which of the following is the logical negation of “Some heffalumps have big ears”?
(1) Some heffalumps do not have big ears.
(2) All heffalumps have big ears.
(3) No heffalumps have big ears.
(4) All of the above.
2. Which of the following is the logical negation of “No heffalumps have forked tongues”?
(1) Some heffalumps do not have forked tongues.
(2) Some heffalumps have forked tongues.
(3) All heffalumps have forked tongues.
(4) No heffalumps do not have forked tongues.
3. Which two of the following are not logical negations of the statement “All heffalumps are gray”?
(1) It is not true that all heffalumps are gray. (2) All heffalumps are not gray.
(3) Not all heffalumps are gray.
(4) Some heffalumps are not gray.
(5) No heffalumps are gray.
4. Write the negation of each of the following.
a. Line  is horizontal.
b. x > 2
d. Some cats are black.
e. No dogs are purple.
c. All pigs are pink.
f. Some cows are not brown.
5. Complete the following:
a. The conjunction p  q is true only if
b. The disjunction p  q is false only if
.
.
6. The statement “x is a multiple of 6 and x is a multiple of 8” is true if x is
(1) 2
(2) 12
(3) 16
(4) 24
7. If the statement “x is an integer or x is an irrational number” is true, which could not be a value of x?

2
(1) –5
(2) 36
(3) 40
(4)
(5)
3
2
8. The statement, “a  5 or b < 2,” is false for which ordered pair (a, b)?
(1) (5, 1)
(2) (5, 2)
(3) (4, 1)
(4) (4, 2)
9. Suppose the statement “Wookiees are furry or Ewoks are scaly” is true. If the statement “Ewoks are scaly”
is false, what can we conclude about Wookiees?
(1) Wookiees are furry.
(2) Wookiees are not furry.
(3) No conclusion is possible.
10. Suppose the statement “Wampas are carnivorous or Banthas have horns” is true. If the statement “Banthas
have horns” is true, what can we conclude about Wampas?
(1) Wampas are carnivorous. (2) Wampas are not carnivorous.
(3) No conclusion is possible.
11. a. If the statement p  q is true and q is false, then p
(1) must be true
(2) must be false
b. If the statement p  q is true and q is true, then p
(1) must be true
(2) must be false
(3) cannot be determined
(3) cannot be determined
Geometry HW: Logic - 2
1. a. What is a conditional statement?
b. What is the hypothesis of a conditional statement?
c. What is the conclusion of a conditional statement?
2. Complete the following: The conditional p  q is false only if
.
3. Suppose the following statements have the given truth values:
Guido does his math homework; true.
Guido watches TV; false.
Guido passes his quiz; true.
Determine whether each of the following is true or false.
a. If Guido does his math homework, then he passes his quiz.
b. If Guido does his math homework, then he watches TV.
c. If Guido watches TV, then he does his math homework.
d. If Guido watches TV, then he does not do his math homework.
e. Guido does his math homework and he watches TV.
f. Guido does his math homework or he watches TV.
g. Guido does his math homework or he passes his quiz.
4. If p  q is true and p  r is false, find the truth value of each of the following:
a. p
b. q
c. r
d. p  q
3. q  r
g. ~q  r
h. (p  r)  r i. (p  r)  r
f. r  p
5. A statement of the form “All ________ verb ________” can be written in the form of a conditional.
For example: “All spiders have eight legs” can be rewritten “If it is a spider, then it has eight legs.”
Note: This is NOT the same as “If it has eight legs, then it is a spider.” (See problems 3b and c above.)
Rewrite each of the following as a conditional (If __________ then _________ .)
a. All squares are rectangles.
b. All isosceles triangles have two congruent sides.
c. All trapezoids have one pair of parallel sides.
6. Norman has just read in his biology text that all frogs are amphibians. Then he goes to the zoo.
a. Norman sees an animal labeled “frog.” He says “Ah hah! That’s an amphibian.” Is he correct?
b. Norman sees an animal labeled “amphibian.” He says “Ah hah! That’s a frog.” Is he correct?
c. Norman sees an animal labeled “NOT a frog.” He says “Ah hah! That’s not an amphibian.” Is he
correct?
d. Norman sees an animal labeled “NOT an amphibian.” He says “Ah hah! That’s not a frog.” Is he
correct?
7. Refer to the truth table for a conditional in your notes to answer the following:
a. If p  q is true and p is true, can we determine the truth value of q? If so, what is it?
b. If p  q is true and q is true, can we determine the truth value of p? If so, what is it?
c. If p  q is true and p is false, can we determine the truth value of q? If so, what is it?
d. If p  q is true and q is false, can we determine the truth value of q? If so, what is it?
Geometry HW: Logic - 3
1. For each of the following pairs of statements, assumed true, give a valid conclusion or, if there is no valid
conclusion, write NC (short for “no conclusion”).
a If it snowed 18 inches on Sunday night, there is no school Monday.
It snowed 18 inches on Sunday night.
b If it snowed 18 inches on Sunday night, there is no school Monday.
There is no school Monday.
c If it snowed 18 inches on Sunday night, there is no school Monday.
It did not snow 18 inches on Sunday night.
d. If it has three eyes, then it is an alien.
Sphynx has three eyes.
e. If Jeremy listens to rap music, his brain turns to mush.
Jeremy doesn’t listen to rap music.
f. If the moon is made of green cheese, space is full of happy mice.
Space is full of happy mice.
g. If Jamie has no homework, she visits her boyfriend.
Jamie has no homework.
h. a  b
i. j  k
j. ~s  t
b
~j
~s
2. For each of the following pairs of statements, assumed true, give a valid conclusion or, if there is no valid
conclusion, write “NC.”
a. Sam goes to the gym or he does his homework.
b. Sam goes to the gym or he does his homework.
Sam does not go to the gym.
Sam goes to the gym.
c. a  b
d. s  ~t
e. ~j  ~k
f. ~y  z
g. x  –2  x  5.
~b
~s
k
~y
x < 5.
3. Tell whether each of the following is a valid argument.
a. If an animal has four legs, then it can run.
b. If an animal has four legs, then it can run.
A horse has four legs.
A dog can run.
A horse can run.
A dog has four legs.
c. If an animal has four legs, then it can run.
A fish does not have four legs.
A fish cannot run.
d. If an animal has four legs, then it can run.
A snake has four legs.
A snake can run.
4. Tell whether each of the following is a valid argument.
a. A dog has teeth or a fish has wings.
b. A dog has teeth or a fish has wings.
A dog has teeth.
A fish does not have wings.
A fish does not have wings.
A dog has teeth.
c. A dog has teeth or a fish has wings.
A dog does not have teeth
A fish has wings.
5. Suppose the statement p  q is true. Then the statement q  p
(1) must also be true. (2) must be false.
(3) could be either true or false.
6. Suppose the statement p  q is true. Then the statement ~p  ~q
(1) must also be true. (2) must be false.
(3) could be either true or false.
Geometry HW: Logic - 4
1. For the statement “If a number is divisible by 4, then it is even,”
a. Give the truth value of the statement.
b. Write the converse of the statement and give its truth value.
c. Write the inverse of the statement and give its truth value.
d. Write the contrapositive of the statement and give its truth value.
2. Write the contrapositive of each statement:
a. If I think, then I exist.
b. If I didn't study, I didn’t pass.
3. Write the converse of the statement, “If Rachel is in ninth grade, then she is taking Geometry”
4. Write the inverse of "If 2x + 5 > 13, then x > 4"
5. Write a conditional that is logically equivalent to “If you cleaned your room, then you watched the movie.”
6. a. Write the inverse of the statement, “If x2  9 then x  3”
b. For what value(s) of x, if any, is the inverse false?
7. Rufus says “If a whole number is divisible by 5, then it ends in five.” Goofus says the converse of Rufus’s
statement. Determine who is right: both of them, Rufus only, Goofus only, or neither. If you believe either
one is wrong, given an example of a number that would make him wrong.
8. Mr. C says “If you don’t learn the vocabulary, then you can’t to the geometry.” Assuming this is a true
statement (it is), then must the statement “If you learn the vocabulary, then you can do geometry” also be
true?
9. Assume the following statements are both true:
If Don watched TV all evening, then he did not pass his math test.
Don passed his math test.
a. Write a conditional that is logically equivalent to the first statement above.
b. Draw a valid conclusion from your answer to part (a) and the second statement above.
10. Explain why the following reasoning is valid: if p  q is true and q is false, then p must be false.
11. For each of the following pairs of true statements, either state a valid conclusion or write “NC.”
a. If a quadrilateral is a rectangle, then it has four right angles.
Quadrilateral PQRS does not have four right angles.
b. If Alexi hit the ball with his hands, then his goal didn’t count.
Alexi’s goal counted.
c. If Lara eats all her vegetable, then she gets ice cream.
Lara gets ice cream.
d. If Kara stops the penalty shot, then her team wins.
Kara stops the penalty shot.
e. If a triangle has base 6 and height 4, then its area is 12.
Triangle ABC does not have base 6 and height 4.
12. Let f represent “Griffins fly” and w represent “Griffins have wings.”
a. Write symbolically “Griffins fly if they have wings.”
b. Write symbolically “Griffins fly only if they have wings.”
c. How are “if” and “only if” related to each other?
Geometry HW: Logic - 5
1. For each pair of statements, assuming they are true, draw a valid conclusion or write “NC.”
a. If a trapezoid is isosceles, then its diagonals are congruent.
Trapezoid ABCD is isosceles.
b. Two lines in a plane intersect or they are parallel.
Lines l and m in plane P are not parallel.
c. If a polygon is a parallelogram, then its interior angles sum to 360.
The interior angles of polygon PART sum to 360.
d. Lamar goes to the movies or he visits his girlfriend.
Lamar goes to the movies.
e. If a parallelogram has congruent diagonals, then it is a rectangle.
If a polygon is a rectangle, then it has four right angles.
f. A polygon is concave or it is convex.
Pentagon PQRST is not convex.
g. a  b
h. c  d
i. c  d
j. r  t
k. v  ~w
l. x  y
m. q  z
a
~d
~d
vr
~v
~x
z
2. James and Allison were trading baseball cards. James hid a card and told Allison, “If this card is a Yankee,
then it’s an All-Star. It’s not an All-Star.” What, if anything, can Allison conclude?
3. If the statement "If Hillary got 100 on the test, then she is happy" is true, what, if anything, can we conclude
if Hillary is not happy?
4. If Fred plays lacrosse, then his dog is named Muttface. Fred's dog is not named Muttface. Which of the
following is a valid conclusion?
(1) Fred plays lacrosse.
(2) Fred does not play lacrosse.
(3) Fred’s dog plays lacrosse.
(4) None of the above.
5. If Sean cleans his room and takes out the trash, then he can watch TV. He took out the trash but he can't
watch TV. What is a logical conclusion?
(1) Sean cleaned his room and took out the trash.
(2) Sean did not clean his room or take out the trash.
(3) Sean cleaned his room and did not take out the trash.
(4) Sean did not clean his room and did take out the trash.
6. Given: If Ellen eats eggplant, she breaks out in purple spots.
If Ellen does not eat eggplant, then her hair does not turn green.
a. What valid conclusion can we draw that does not involve eggplant?
b. How can we tell just by looking at Ellen whether she ate eggplant? Explain your answer.
7. Let p represent “Lisa passed the test” and s represent “Lisa studied.”
a. Write symbolically “Lisa passed the test if she studied.”
b. Write symbolically “Lisa passed the test only if she studied.”
c. The conjunction “Lisa passed the test if she studied and Lisa passed the test only if she studied” is called
a biconditional and is usually shorted to “Lisa passed the test if and only if she studied.” In logic, it is
written p  s. The statement p  s is true only if p  s and s  p are both true. For what values of p
and s will p  s bed true?
Geometry HW: Logic - 6
1. When is a biconditional true?
2. Give the truth value of each of the following Biconditionals.
a. Betty Boop (an NNCS high school student) passes the test if and only if she gets 65 or above.
b. The earth is flat if and only if pigs have wings.
c. A person lives in New York State if and only if (s)he lives in the United States.
d. 3x – 4 < 11 if and only if x < 5.
e. x2 = 25 if and only if x = 5.
3. Fact: A rhombus is a square if and only if its diagonals are congruent.
Using the above fact, draw a valid conclusion about each of the following rhombi.
a. Rhombus ABCD has congruent diagonals.
b. Rhombus EFGH is a square.
c. Rhombus JKLM does not have congruent diagonals.
d. Rhombus PQRS is not a square.
4. Write each of the following definitions as two separate conditionals.
a. Isosceles triangles have two congruent sides.
b. Perpendicular lines form right angles.
5. Tell whether each of the following is a good definition. If not, explain why not.
a. An equilateral triangle is a triangle with all three sides congruent.
b. A square is a quadrilateral with all four sides congruent.
c. A vertebrate is an animal having a backbone.
d. A bird is an animal that lays eggs.
6. Definition: Three or more points are collinear if they all lie on the same line.
a. Draw a diagram showing that points A, B and C are collinear.
b. Draw a diagram showing that points D, E and F are not collinear.
7. Draw a valid conclusion from each pair of givens or write “NC.”
a. a  b
b. c  d
c. c  d
d. r  t
e. v  ~w
~a
~d
c
t
w
f. x  y
x
g. q  z
kq
Geometry HW: Logic - 7
For #1 – 4, write a complete logic proof in statement-reason form.
1. Given:
If Fred eats worms, then he will get sick.
If Fred gets sick, then he will miss school.
Fred ate worms.
Let W: Fred eats worms.
S: Fred gets sick.
M: Fred misses school.
Use the givens above and logical reasoning to determine if Fred is in school.
2. Given:
If Jack goes to the dance, he does not study for his test.
Jack goes to the school dance or Jill is angry.
Jack studies for the test.
Let D : Jack goes to the dance.
S : Jack studies for the test. A : "Jill is angry,"
Use the givens above and logical reasoning to determine if Jill is angry.
3. Given:
If the Mad Hatter doesn't have a tea party, then Alice doesn't answer the riddle.
If the Mad Hatter has a tea party, then the Dormouse sleeps.
Alice answers the riddle or the Queen commands "Off with her head!"
The Dormouse doesn't sleep.
Let P: The Mad Hatter has a tea party.
R: Alice answers the riddle.
S: The Dormouse sleeps.
Q: The Queen commands “Off with her head!”
Use the givens above and logical reasoning to determine if the Queen commands "Off with her head!"
4. (With thanks to Lewis Carroll.)
Given: If I do not grumble while doing a homework problem, then I understand the
problem.
This problem is not set up like ones I am used to.
If the problem is easy, it does not give me a headache.
If a problem is not set up like ones I am used to, I do not understand it.
If I grumble while doing a homework problem, then I have a headache.
Let G: I grumble while doing a HW problem. U: I understand the problem.
E: The problem is easy.
S: The problem is set up like ones I’m used to.
H: the problem gives me a headache.
Use the givens above and logical reasoning to determine if this problem is easy.
5. Two high school students, Rufus and Doofus, are talking to each other on cheap cell phones.
a. Rufus hears Doofus say “If (loud burst of static), then I’m the Queen of Sheba.” Doofus’s statement
(1) must have been true.
(2) must have been false.
(3) could have been either true or false.
b. Doofus hears Rufus reply “If you’re the Queen of Sheba, then (another loud burst of static).” Rufus’s
statement
(1) must have been true.
(2) must have been false.
(3) could have been either true or false.
c. Rufus hears Doofus say “If (loud burst of static), then we have bad reception.” Doofus’s statement
(1) must have been true.
(2) must have been false.
(3) could have been either true or false.
d. Doofus hears Rufus say “If this phone makes one more burst of static, then (loud burst of static).”
Rufus’s statement
(1) must have been true.
(2) must have been false.
(3) could have been either true or false.
Geometry HW: Logic - 8
Write complete proofs, in statement-reason form, for the following.
1. Given: If Lara is hungry, then she eats.
If Lara is not hungry, then she throws her food.
Lara is not throwing her food.
Let H represent "Lara is hungry."
E represent "Lara eats."
T represent "Lara throws her food."
Using the laws of logic, prove that Lara is eating.
2. Given: The Cheshire Cat smiles or he disappears.
If Alice talks to the Cat, then the Queen won't threaten to cut off everyone's head.
If the Cheshire Cat smiles, then Alice talks to him.
The Queen threatens to cut off everyone's head.
Let S represent "The Cheshire Cat smiles."
D represent "The Cheshire Cat disappears."
T represent "Alice talks to the Cat."
Q represent "The Queen threatens to cut off everyone's head."
Using the laws of logic, prove that the Cheshire Cat disappears.
3. Given: You get to work at nothin' all day or you're takin' care of business.
If you catch the 8:15 into the city and your train is on time, you get to work by 9:00.
If you get to work by 9:00, you don't get annoyed.
If you don't catch the 8:15 into the city, you are self-employed.
If you are self-employed, you don't get to work at nothin' all day.
You are annoyed.
Your train was on time.
Let A represent "You get annoyed."
N represent "You get to work at nothin' all day."
B represent "You're takin' care of business."
T represent "Your train was on time."
S represent "You're self-employed."
C represent "You catch the 8:15 into the city."
W represent "You get to work by 9:00."
Using the laws of logic, prove that you are takin' care of business.
Geometry HW: Logic - Review
1. a. What is the rule for AND?
b. What is the rule for OR?
c. What is the rule for IF . . . THEN . . . ?
2. If the statement, “x is an integer or x is an irrational number,” is true, which could not be a value of x?
2

(1) 25
(2) 40
(3)
(4)
(5) 0
3
3
3. If the statement "If Matt wins $100 in the lottery, then he is happy" is true, which must be true if Matt is not
happy?
(1) Matt did not win any money in the lottery.
(2) Matt won $100 in the lottery.
(3) Matt did not win $100 in the lottery.
(4) Matt won more than $100 in the lottery.
4. Mary says, “If the number I’m thinking of is a multiple of 12, then it is greater than 12.” Mary’s statement
is false if the number she is thinking of is
(1) 6
(2) 12
(3) 18
(4) 24
5. The statement, “a > 4 and b  4,” is true for which ordered pair (a, b)?
(1) (4, 4)
(2) (4, 5)
(3) (5, 5)
(4) (5, 4)
6. What is the converse of the statement, “If Rachel is in the ninth grade, she is taking Geometry?”
(1) If Rachel is taking Geometry, she is in the ninth grade.
(2) If Rachel is not taking Geometry, she is not in the ninth grade.
(3) If Rachel is not in the ninth grade, she is taking Geometry.
(4) If Rachel is not in the ninth grade, she is not taking Geometry.
7. The statement, “If two triangles are congruent, then they have equal areas,” is
(1) true, and the converse of the statement is true.
(2) true, and the converse of the statement is false.
(3) false, and the converse of the statement is true.
(4) false, and the converse of the statement is false.
8. Which statement is the inverse of the statement, “If Amy studies, she will pass the exam”?
(1) If Amy does not pass the exam, she has not studied.
(2) If Amy passes the exam, then she has studied.
(3) If Amy does not study, she will not pass the exam.
(4) Amy passes the exam if she studies.
9. Which statement is logically equivalent to “If Mindy was quiet, then she was not defenestrated.”
(1) Mindy was quiet and she was not defenestrated.
(2) If Mindy was not quiet, then she was defenestrated.
(3) If Mindy was not defenestrated, then she was quiet.
(4) If Mindy was defenestrated, then she was not quiet.
(This assignment is continued on the next page.)
10. Mary got Larry a Christmas present. Mary tells Larry, “If your present is edible, then it’s chocolate.” Harry
says “It’s not chocolate.” If both Mary and Harry are telling the truth, what can Larry conclude about his
present?
(1) It’s edible.
(2) It’s not edible.
(3) There is nothing to conclude.
(4) What other choice could there be?
11. Write the following as two separate conditionals:
a. A parabola y = ax2 + bx + c opens up if and only if a > 0.
b. A rhombus is a quadrilateral with all four sides congruent.
12. The statement “If a quadrilateral is a rectangle, then it has congruent diagonals” is true. Write each of the
following related conditionals and give its truth value. If any are false, illustrate with a diagram.
a. Converse
b. Inverse
c. Contrapositive
13. Consider the following three statements: 7 is a lucky number or 13 is a lucky number. If 13 is not a lucky
number, then problem #13 is confusing. Problem #13 is confusing. If these statements are true, which of
the following is the best conclusion?
(1) Only 7 is a lucky number.
(2) Only 13 is a lucky number.
(3) Both 7 and 13 are lucky numbers. (4) Neither 7 nor 13 is a lucky number.
(5) None of these
14. Solve for x and graph the solution set on a number line: 2x  5  9 and 2x  5  9 .
15. Solve for x and graph the solution set on a number line: 3x  2  5 or 3x  2  5 .
16. For each pair of statements, assuming they are true, draw a valid conclusion or write “NC.”
a. If a number is composite, then it has more than two factors.
The number 289 is composite.
b. Quadrilateral ABCD is a trapezoid or it is a rectangle.
Quadrilateral ABCD is not a rectangle.
c. If a quadrilateral is a rectangle, then it is a parallelogram.
Quadrilateral PQRS is not a parallelogram.
d. Edith does Geometry homework or she watches TV.
Edith watches TV.
e. If imps limp, then gnomes roam.
If gnomes roam, then sprites fight.
f. c  d
g. r  s
h. p  q
i. a  b
j. v  ~w
k. x  y
l. a  b
~d
~s
~p
bc
v
x
b
17. There is no proof on this review. There is a proof on the test. If you have already forgotten how to do logic
proofs, review assignments 7 and 8.
Review Answers
1a. Only T if both parts T b. Only F if both parts F (T if either part T). c. Only F when T  F.
2. (3)
3. (3)
4. (2)
5. (4)
6. (1)
7. (2)
8. (3)
9. (4)
10. (2)
11a. If a parabola y = ax2 + bx + c opens up, the a > 0. If a > 0, then the parabola y = ax2 + bx + c opens up.
b. If a quadrilateral is a rhombus, then all four sides are congruent.
If a quadrilateral has all four sides congruent, then it is a rhombus.
12a. If a quadrilateral has congruent diagonals , then it is a rectangle. False.
b. If a quadrilateral is a not rectangle, then it does not have congruent diagonals. False.
c. If a quadrilateral does not have congruent diagonals, then it is not a rectangle. True.
13. (5)
16a. The number 289 has more than two factors. b. Quadrilateral ABCD is a trapezoid.
c. Quadrilateral PQRS is not a rectangle.
d. NC
e. If imps limp, then sprite fight.
f. ~c
g. r
h. NC
i. a  c
j. NC
k. y
l. NC
STUFF YOU SHOULD KNOW:
Vocabulary
negation
conjunction
disjunction
conditional
biconditional
converse
inverse
contrapositive
logically equivalent
definition
Rules for Connectives
Not/negation
And
Or
If . . . then
. . . if and only if . . .
Laws of Reasoning (The rules, not the names)
Law of Detachment
Chain Rule
Law of the Contrapositive
Law of Modus Tollens
Law of Disjunctive Inference
How to:
Find the truth value of a compound statement
Negate statements of the form “All . . . ,” “Some . . . ,” or “No . . . “
Draw a logically valid conclusion from givens
Write a statement logically equivalent to a given conditional