Geometry Chapter 1 Review

Geometry Chapter 1 Review

Mr. Baker's Geometry

​What to Expect in This Review:

Each section of the review will have a summary slide over the lesson. Following that will be 3-5 questions based on that lesson.

​

​The Chapter 1 Test will test you on:

• ​Knowledge of vocabulary

• ​Analysis of logical reasoning

• ​Applying logic

​Lesson 1.1 - Inductive Reasoning

Inductive reasoning is a type of logical reasoning that looks at patterns  to form a conjecture.

​

​Example 1: What number goes next?

4, 5, 7, 10, 14, ?

The pattern is increasing the sum to the next number in the sequence. 4 + 1 = 5, 5 + 2 = 7, 7 + 3 = 10, 10 + 4 = 14.

​

​So 14 + 5 = 19. The next number is 19.

Example 2: How many dots are in the 4th step?

​Step 1: 1 row of 8 dots = 8 total

​Step 2: 2 rows of 9 dots = 18 total

​Step 3: 3 rows of 10 dots = 30 total

​Step 4 will have 4 rows of 11 = 44 total

Fill in the Blank

Practice Problem 1

What number goes next in the sequence?

1/48, 1/24, 1/12, 1/6, _

Fill in the Blank

Practice Problem 2

What number is missing in the sequence?

1, 3, 6, 10, 15, 21, _, 36

Fill in the Blank

Practice Problem 3

How many dots are in step 4?

Fill in the Blank

Practice Problem 4

How many dots are in the 14th step?

Multiple Choice

Practice Problem 5

What is inductive reasoning?

• A type of logical reasoning that looks at patterns  to form a conjecture

• A type of logical reasoning that uses rules, definitions, or properties to reach logical conclusions from given statements

• A type of deductive reasoning which states that if p→q is true and p is true, then q is true.

• A type of deductive reasoning which states that if p→q and q→r, then p→r.

​Lesson 1.2 - Venn and Euler Diagrams

A Venn Diagram is a diagram that shows all possible relationships that can exist between data sets.

​​The above diagram shows that all living four-legged creatures are animals, but not minerals. There is no overlap because no creature is both an animal and a mineral.

A Euler Diagram is a diagram that shows all relationships that exist in actuality.

The above diagram shows all possible ways students can take three different classes.

​Lesson 1.2 - Venn and Euler Diagrams

​1. How many students take both Spanish and Chemistry?

​18 students. Spanish and Chemistry overlap over the 15 and 3. So 18 people take both Spanish and Chemistry.

​

​2. How many take Math or Chemistry?

​185 students. Add together all of the students in the math circle and chemistry circle. 85+5+60+15+17+3 = 185.

Fill in the Blank

Practice Problem 6

How many students take Spanish, Chemistry, and Math?

Fill in the Blank

Practice Problem 7

How many students take none of the classes?

Fill in the Blank

Practice Problem 8

How many students take Chemistry and Math, but not Spanish?

Multiple Choice

Practice Problem 9

What is a Venn Diagram?

• A diagram that shows all possible relationships that can exist between data sets.

• A diagram that shows all relationships that exist in actuality.

• A diagram that shows the relationship between the independent and dependent variable.

• A diagram that maps domain to range.

Multiple Choice

Practice Problem 10

What is an Euler Diagram?

• A diagram that shows all possible relationships that can exist between data sets.

• A diagram that shows all relationships that exist in actuality.

• A diagram that shows the relationship between the independent and dependent variable.

• A diagram that maps domain to range.

​Lesson 1.3 - Correlation vs. Causation

Two data sets that measure two different variables can sometimes have a correlation, or a defined relationship.

• ​Positive correlation: as the value of one variable increases, the value of the other increases.

• ​Negative correlation: as the value of one variable increases, the value of the other decreases.

• ​No correlation: No clear relationship exists between the two variables.​

​

Causation occurs when one variable directly contributes to the change of the other variable.

​

​**Not all correlation = causation. A lurking variable, or a related cause not measured, may contribute to the change of both variables.

Multiple Choice

Practice Problem 111

Which example shows correlation, but not necessarily causation?

• I put water in the freezer and now I have ice.

• The alarm clock went off and I woke up.

• Temperature and the amount of people at the beach.

Multiple Choice

Practice Problem 12

Which example shows a causation?

• High rate of social media use and reduced grades in school.

• My car ran out of gas and now I am stranded on the side of the road.

• Recess in elementary students and classroom behavior.

Multiple Choice

Practice Problem 13

Which type of correlation is represented? "The number of public service campaigns for drivers to wear seatbelts and the number of car accident deaths."

• Positive correlation

• Negative correlation

• No correlation

• Linear correlation

Multiple Choice

Practice Problem 14

Which could possibly be a lurking variable for the scenario described?

Diet soda sales vs. number of car accidents

• People are distracted while drinking diet soda and driving, so more accidents happen when drinking diet soda.

• Weight loss. More people could be losing weight and not driving.

• A higher population. More people means more diet soda sold and more accidents. It's just numbers. 

• Models of cars. Some cars are more prone to be in accidents.

Multiple Choice

Practice Problem 15

What is correlation?

•  A defined relationship between two data sets

• A relationship where one variable directly causes a change in the other

• A variable not measured

​Lesson 1.4 - Logic

A statement p is either true or false. These values can be manipulated through logical operations.

• ​Negation: Adding the word "not." Symbolized as ~p.

• ​Conjunction: Adding the word "and" to two statements. Symbolized as p /\ q.

• ​Disjunction: Adding the word "or" to two statements. Symbolized as p \/ q.

​Examples:

​p: Christmas is on December 25. q: The state of California borders the state of Maine.

​

​~p is FALSE, because p is initially true.

​p/\q is FALSE, because statement q is false.

​p\/q is TRUE, because at least one statement (in this case p) is true.

Multiple Choice

Practice Question 16

p: Tyler has the flu.

q: Renee quit the softball team.

Which statement is p\/q?

• Tyler does not have the flu.

• Renee did not quit the softball team.

• Tyler has the flu and Renee quit the softball team.

• Tyler has the flu or Renee quit the softball team.

Multiple Choice

Practice Question 17

p: Enrique's mother is from Puerto Rico.

q: Alicia's gerbil ate her car keys.

Which statement is ~p/\q?

• Enrique's mother is not from Puerto Rico and Alicia's gerbil ate her car keys.

• Enrique's mother is not from Puerto Rico or Alicia's gerbil ate her car keys.

• Enrique's mother is from Puerto Rico and Alicia's gerbil did not eat her car keys.

• Enrique's mother is from Puerto Rico or Alicia's gerbil did not eat her car keys.

Multiple Choice

Practice Question 18

p: The new movie does not come out next Friday

q: Shawn turned in his assignment late.

Which statement is ~p/\~q?

• The new movie does not come out next Friday and Shawn did not turn in his assignment late.

• The new movie comes out next Friday and Shawn did not turn in his assignment late.

• The new movie comes out next Friday or Shawn did not turn in his assignment late.

• The new movie does not come out next Friday or Shawn turned in his assignment late.

​Lesson 1.5 - Truth Tables

A truth table is a table that maps out all possible combinations of truth values of statements.

​

​Below is the truth table of ~p \/ q:

Multiple Choice

Practice Question 19

For the truth table of p/\~q, what truth value goes in the ?

• True

• False

Multiple Choice

Practice Question 20

For the truth table of ~(~p\/q), what truth value goes in the ?

• True

• False

Multiple Choice

Practice Question 21

p is true and q is true. What is the truth value of ~p/\~q?

• True

• False

Multiple Choice

Practice Question 22

p is false, q is false, and r is true. What is the truth value of ~(p\/q) /\ ~r?

• True

• False

​Lesson 1.6 - Conditional Statements

A conditional statement is an operation on two statements p and q that creates the condition "if p, then q." This is symbolically said as p→q.

• ​Statement p is called the hypothesis and statement q is called the conclusion.

​p: A number is a whole number

​q: A number is an integer.

​p→q: If a number is a whole number, then it is an integer.

Truth table of conditional statements

​Lesson 1.6 - Conditional Statements

Related conditionals are other statements that can be formed from a conditional statement.

• ​Converse: Exchanging the hypothesis and conclusion (q→p).

• ​Inverse: Negating the hypothesis and conclusion (~p→~q)

• ​Contrapositive: Exchanging and negating the hypothesis and conclusion (~q→~p).

​Example:      p: A number is a whole number             q: A number is an integer.

​Conditional Statement p→q: If a number is a whole number, then it is an integer.

​Converse q→p: If a number is an integer, then it is a whole number.

​Inverse ~p→~q: If a number is not a whole number, then it is not an integer.

​Contrapositive ~q→~p: If a number is not an integer, then it is not a whole number.

Multiple Choice

Practice Problem 23

What is the hypothesis of the conditional?

"If the product of two numbers is 1, then the two numbers are multiplicative inverses."

• The product of two numbers is 1

• The two numbers are multiplicative inverses

• The product of two numbers is not 1

• The two numbers are not multiplicative inverses

Multiple Choice

Practice Problem 24

Rewrite the following as a conditional statement:

"Get a month free after signing up for one-year subscription."

• If you get a month free, then you signed up for a one-year subscription.

• If you signed up for a one-year subscription, then you get a month free.

• If you don't sign up for a one-year subscription, then you won't get a month free.

• If you don't get a month free, then you didn't sign up for a one-year subscription.

Multiple Choice

Practice Problem 25

What is the converse of the statement below?

"If you order in the next 5 minutes, then you'll get 15% off of the price."

• If you don't get 15% off of the price, then you don't order in the next 5 minutes.

• If you don't order in the next five minutes, then you won't get 15% off of the price.

• If you get 15% off of the price, then you'll order in the next 5 minutes.

Multiple Choice

Practice Problem 26

What is the contrapositive of the following statement?

If you buy a small popcorn, then you can't get free drink refills.

• If you can't get free drink refills, then you bought a small popcorn.

• If you get free drink refills, then you didn't buy a small popcorn.

• If you don't buy a small popcorn, then you can get free drink refills.

Multiple Choice

Practice Problem 27

What is the inverse of a conditional statement p→q?

• p→q

• q→p

• ~p→~q

• ~q→~p

​Lesson 1.7 - Biconditional Statements

Conditional statements and their related conditionals are related in their truth values.

• ​A conditional statement and its contrapositive are logically equivalent--both have the same truth value.

• ​A converse and inverse are logically equivalent--both have the same truth value.

Example:

​Conditional statement: "If a number is divisible by 4, then it is even."

• ​This statement is true. ∴ The contrapositive is automatically true.

​Its converse is "If a number is even, then it is divisible by 4."

• ​This statement is false (a counterexample is 10). ∴The inverse is automatically false.

​Lesson 1.7 - Biconditional Statements

If a conditional and its converse are both true, then this statement can be written as a biconditional statement. A biconditional statement is the conjunction of a true conditional and its true converse. This is written symbolically as p↔q and the sentence uses the phrase "if and only if," which can be abbreviated as iff.

​*False biconditional statements cannot be written.

Example:

True conditional statement: "If a polygon has 4 sides, then it is a quadrilateral."

​True converse: "If a polygon is a quadrilateral, then it has 4 sides."

Biconditional statement: "A polygon has 4 sides if and only if it is a quadrilateral"

• ​It can also be written as "A polygon is a quadrilateral iff it has 4 sides."

Multiple Choice

Practice Problem 28

Is the contrapositive of the following statement true or false?

"If a person is at least 16 years of age, then they can apply for a drivers license."

• True

• False

Multiple Choice

Practice Problem 29

What is the truth value of the inverse of the following statement?

"If 75 is divisible by 5, then it is divisible by 25.

• True

• False

Multiple Select

Practice Problem 30

Select ALL true biconditional statements.

• An angle is obtuse iff its measure is greater than 90 degrees.

• An angle's measure is greater than 90 degrees if and only if its an obtuse angle.

• A number is rational iff it can be written as a fraction.

• A number can be written as a fraction if and only if it is a whole number.

Multiple Select

Practice Problem 31

What are true statements that can be derived from the biconditional statement "ab = 1 iff a and b are reciprocals"?

• If ab = 1, then a and b are reciprocals.

• If a and b are reciprocals, then ab = 1.

• If ab ≠ 1, then a and b are reciprocals.

• If ab = 1, then a and b are not reciprocals.

​Lesson 1.8 - Deductive Reasoning 1:

The Law of Detachment

Deductive reasoning is A type of logical reasoning that uses rules, definitions, or properties to reach logical conclusions from given statements.

​

Example:

​If John is late making his car insurance payment, he will be assessed a late fee of $50. John’s payment is late this month, so he knows that he will be assessed a late fee of $50.

​

​This is deductive reasoning because John knows what the consequences of late payments are based on the policy.

​Lesson 1.8 - Deductive Reasoning 1:

The Law of Detachment

The Law of Detachment is a type of deductive reasoning which states that if p→q is true and p is true, then q is true.

​Example:

​Given: If a number is a rational number, then it is a real number.

​7/3 is a rational number.

• ​By the Law of Detachment, because 7/3 is a rational number, we can conclude from the given statement that it is also a real number.

• ​

∴ 7/3 is a real number.

Multiple Choice

Practice Problem 32

Which of the following is a valid conclusion for the following statements?

Given: If a person is caught speeding, then they will get a ticket.

Trevor was caught speeding.

• ∴ Trevor will get in trouble.

• ∴ Trevor was caught speeding.

• ∴ Trevor will get a ticket.

• We cannot make a valid conclusion because we cannot apply the Law of Detachment

Multiple Choice

Practice Problem 33

Which of the following is a valid conclusion for the following statements?

Given: If x < 5, then x < 9.

x = 0.

• ∴ x < 5.

• ∴ x < 9.

• ∴ x = 0.

• We cannot make a valid conclusion because we cannot apply the Law of Detachment

Multiple Choice

Practice Problem 34

What is a valid conclusion to the following statement?

Given: If a person runs for 30 minutes, then they will burn 200 calories.

Kyla burned 200 calories.

• ∴ Kyla ran for 30 minutes.

• ∴ Kyla exercised.

• ∴ Kyla started to diet.

• We cannot make a valid conclusion because we cannot apply the Law of Detachment

Multiple Select

Practice Problem 35

Select ALL reasons that the conclusion is invalid.

Given: If the class raises $1000, then the class will have a party.

The class has a party.

Conclusion: ∴ The class raised $1000.

• The given is p→q. The given information corresponds to q. Therefore, we cannot conclude that p happened.

• The class could have a party for other reasons.

• The class raised $1000, but then could have decided not to have the party.

• The conclusion is valid.

Multiple Choice

Practice Problem 36

What is deductive reasoning?

• A type of logical reasoning that looks at patterns  to form a conjecture

• A type of logical reasoning that uses rules, definitions, or properties to reach logical conclusions from given statements

• A type of deductive reasoning which states that if p→q is true and p is true, then q is true.

• A type of deductive reasoning which states that if p→q and q→r, then p→r.

Multiple Choice

Practice Problem 37

What is the Law of Detachment?

• A type of logical reasoning that looks at patterns  to form a conjecture

• A type of logical reasoning that uses rules, definitions, or properties to reach logical conclusions from given statements

• A type of deductive reasoning which states that if p→q is true and p is true, then q is true.

• A type of deductive reasoning which states that if p→q and q→r, then p→r.

​Lesson 1.9 - Deductive Reasoning 2:

The Law of Syllogism

The Law of Syllogism is a type of deductive reasoning which states that if p→q and q→r, then p→r.

​Example:

​Given: If a hurricane is Category 5, then winds are greater than 155 miles per hour.

If winds are greater than 155 miles per hour, then trees, shrubs, and signs are blown down.

​

​Conclusion: ∴ If a hurricane is Category 5, then trees, shrubs, and signs are blown down.

Multiple Choice

Practice Problem 38

What is a valid conclusion to the following statement?

If a number is a whole number, then it is an integer. If a number is an integer, then it is a rational number.

• ∴ If a number is a rational number, then it is a whole number.

• ∴ If a number is a whole number, then it is a rational number.

• ∴ A number is a rational number.

• We cannot make a valid conclusion because we cannot apply the Law of Syllogism

Multiple Choice

Practice Problem 39

What is a valid conclusion to the following statement?

If an organism is a parasite, then it survives by living on or in a host organism. If a parasite lives in or on a host organism, then it harms its host.

• ∴ If an organism is a parasite, then it survives by living on or in a host organism.

• ∴ If an organism's host is harmed, then the organism is a parasite.

• ∴ If an organism is a parasite, then it harms its host.

• We cannot make a valid conclusion because we cannot apply the Law of Syllogism

Multiple Choice

Practice Problem 40

What is a valid conclusion to the following statement?

If a number is a squared number, then it is divisible by 4. If a number is a squared number, then its square root is a whole number.

• ∴ If a number is divisible by 4, then its square root is a whole number.

• ∴ If a number is divisible by 4, then it is a squared number.

• ∴ If a number's square root is a whole number, then it is a squared number.

• We cannot make a valid conclusion because we cannot apply the Law of Syllogism

Multiple Choice

Practice Problem 41

What is the Law of Syllogism?

• A type of logical reasoning that looks at patterns  to form a conjecture

• A type of logical reasoning that uses rules, definitions, or properties to reach logical conclusions from given statements

• A type of deductive reasoning which states that if p→q is true and p is true, then q is true.

• A type of deductive reasoning which states that if p→q and q→r, then p→r.

Poll

What did you think of this type of review session?

• Very helpful

• Somewhat helpful

• A little helpful

• Not helpful

Open Ended

Is there anything you would change about this type of guided review?