{{ 'ml-label-loading-course' | message }}
{{ toc.name }}
{{ toc.signature }}
{{ tocHeader }} {{ 'ml-btn-view-details' | message }}
{{ tocSubheader }}
{{ 'ml-toc-proceed-mlc' | message }}
{{ 'ml-toc-proceed-tbs' | message }}
Lesson
Exercises
Recommended
Tests
An error ocurred, try again later!
Chapter {{ article.chapter.number }}
{{ article.number }}. 

{{ article.displayTitle }}

{{ article.intro.summary }}
Show less Show more expand_more
{{ ability.description }} {{ ability.displayTitle }}
Lesson Settings & Tools
{{ 'ml-lesson-number-slides' | message : article.intro.bblockCount }}
{{ 'ml-lesson-number-exercises' | message : article.intro.exerciseCount }}
{{ 'ml-lesson-time-estimation' | message }}
When police detectives want to solve a crime, they scan the crime scene for clues. They develop possible scenarios based on what they find and then focus on the scenario that seems most likely based on these limited observations. This type of analytical thinking is known as inductive reasoning. Conclusions are drawn from an incomplete picture, so they may be incorrect.
Detective

Another type of analytical thinking is deductive reasoning. In deductive reasoning, claims are supported by facts and are connected to consequent facts. Coming to a conclusion is like putting together the pieces of a jigsaw puzzle. Since all of the pieces fit neatly together, conclusions drawn by this type of reasoning tend to be stronger than those drawn by inductive reasoning.

jigsaw puzzle

For example, if a house was robbed and there is only one forced door, it is a fact that the thief entered through that door. Interested in knowing more about these types of reasoning? Stay and enjoy!

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Explore

Which Is More Reliable?

Kevin loves movies and television shows where mysteries are solved. When he grows up he would like to be a detective πŸ•΅πŸΎβ€β™‚οΈ. He is always helping his friends to clear their doubts. Last week, Dylan and Maya came to him for help.
Kevin talking to Dylan and Maya
In what situation is Kevin's advice based on observations? In what situation is Kevin's advice based on facts? Which of Kevin's tips is more reliable?
Discussion

Looking for Patterns

When not all the information about a particular situation is known, the existing information can be analyzed to look for some kind of pattern. If such a pattern exists, it can lead to some conclusions. This way of thinking is known as inductive reasoning.

Concept

Inductive Reasoning

Inductive reasoning is the process of finding patterns in specific observations and writing a conclusion or conjecture. Since the conjecture is based on observations, it might be false. For example, suppose an observer notices that all the birds around them are white. The observer might inductively reason that all birds in the world are white, which is not true.
Man watching four white doves, then makes a conjecture. Look and behold, he sees a red bird!
Inductive reasoning is a practical method used in geometry and algebra for recognizing visual and numerical patterns. For instance, use the first three figures in the following diagram to guess the shape and the number of cubes in the fourth figure.
Four figures following a pattern.
After observing the first three figures, it is reasonable to conclude that the number of cubes increases by from one figure to the next — one on the top, one on the front, and one on the right. Since the figure starts with cube and each step adds cubes, an expression can be written to model the number of cubes in each figure.
Here, represents the number of steps after Figure For example, Figure is one step after Figure so This expression allows the calculation of the number of cubes in any figure of the pattern.
Number of Cubes
Figure
Figure
Figure
Figure
Figure
Example

The Thief's Pattern

It is Saturday night and Kevin, his parents, and his sister are gathered in the living room watching an action movie. The movie is about a thief who leaves riddles for the police at every crime scene. The police have collected the following cards from the first four robberies.

Four triangles in a pattern. First is one triangle. Second is four triangles of same size as first. Third and fourth are nine and sixteen triangles respectively.

Each card has a riddle written on the back that the police must solve to determine where the next robbery will take place.

a Determine how many green triangles will be on the card that the thief will leave after his next hit.
b The thief wrote the following riddle on the back of the fourth card.
What is the number of green triangles in the seventeenth card?
What is the number of the house that the gang plans to rob?

Hint

a Use inductive reasoning. Start by counting the number of green triangles on each card. Notice that card has two more green triangles than Card and Card has three more green triangles than Card
b Since the seventeenth card is quite far into the pattern, try to find an algebraic expression representing the number of green triangles on each card. Notice that Card has green triangles and Card has green triangles.

Solution

a It is asked to find the number of green triangles on the fifth card. This can be found by observing the first four cards in the pattern — in other words, finding the answer requires using inductive reasoning. Start by counting the number of green triangles in each of the cards.
A pattern of figures
Write the results in a table to organize the information. Make a column for the number of the card and a column for the number of green triangles on the card.
Card Green Triangles

The next step is trying to find a relation between the numbers in the table. Comparing the rows, the following relations can be written.

  • Card has more triangles than Card
  • Card has more triangles than Card
  • Card has more triangles than Card
If this pattern continues, then Card would have more triangles than Card Therefore, the fifth card will have green triangles.
A pattern of figures
b Since the card is quite far into the pattern, it would take a while to draw the diagram for the figure. The police do not have much time to prevent the next robbery. Use the table made in Part A to find an algebraic expression that represents the number of green triangles on each card.
Card Green Triangles

Card has green triangles, which can be written as Similarly, Card has green triangles, which can be written The same happens with Card

Card Green Triangles
If this pattern continues, the number of green triangles in the card is equal to the sum of the first natural numbers.
Consequently, Card will have green triangles. With this information, the police now have the complete address of the next robbery!
All units to House 153 on Boulevard Avenue. Now!
Discussion

Logical Reasoning

Another way of analyzing facts is through deductive reasoning. Unlike inductive reasoning, deductive reasoning does not rely on approximations or guesses. It is a process of logical reasoning through which two or more pieces of information are combined to arrive at a conclusion.

Concept

Deductive Reasoning

Deductive reasoning is the process of reaching logical conclusions from given statements. As long as the given statements are true, the conclusions drawn using deductive reasoning are also true. Consider the following two statements.
Two statements combined to obtain a conclusion
The conclusion is true because the given statements are true. Unlike inductive reasoning, deductive reasoning uses facts, definitions, properties, and the laws of logic to reach a conclusion. The two main laws of logic used to form a conclusion in deductive reasoning are the Law of Detachment and the Law of Syllogism.
Example

An Odd Riddle

The police arrived at house on Boulevard Avenue too late and the gang got away. At the new crime scene, the police found the fifth card, which had the following riddle.

The sum of two odd numbers is odd or even?
Determine the answer to the thief's question using both inductive and deductive reasoning.

Answer

See solution.

Hint

Inductive reasoning is the process of finding patterns in specific observations and writing a conclusion. Find the sum of random odd numbers and analyze the results. Deductive reasoning is the process of reaching logical conclusions from given statements. Any odd number can be written as with an integer number.

Solution

Start by recalling what inductive and deductive reasoning are.

Definition
Inductive Reasoning The process of finding patterns in specific observations and writing a conclusion
Deductive Reasoning The process of reaching logical conclusions from given statements

Keeping the definitions in mind, the answer will be found by applying each reasoning procedure.

Using Inductive Reasoning

The thief's question involves the sum of two odd numbers. Think of two arbitrary odd numbers and calculate their sum.

Odd Numbers Sum

Take a look at the numbers in the right-hand side column. All of the numbers are even, so it is natural to say that the sum of two odd numbers is always even.

The sum of two odd numbers is an even number.

Note that this conclusion is based on the pattern observed in the table. In other words, there is no certainty that the conclusion is true for any two odd numbers. This is not good news to the detective.

Using Deductive Reasoning

Considering random odd numbers does not qualify as deductive reasoning because this type of reasoning is not based on guesses. Instead, consider two odd numbers written in a general form. Recall that any odd number can be written as where is an integer number. Let and be two different odd numbers.
Since the purpose is to study the sum of two odd numbers, add and and simplify as much as possible.
Notice that the sum of and is equal to times Since and are integers, the sum inside the parentheses is an integer. Then, the sum of the two odd numbers has the form where
This form matches the general form of even numbers. Therefore, is an even number.

The sum of two odd numbers is an even number.

Conclusion

As shown, both conclusions are the same but were found using different types of reasoning.
However, deductive reasoning is the most formal way to solve the question because it shows that the sum is an even number, no matter what odd numbers are chosen. Understanding this and fed up with being played for a fool by the gang, the detective put all his efforts into catching them and discovered the next target very quickly.
Discussion

Unproven Statements

Mathematicians often make claims after detecting patterns. However, these claims cannot be accepted as fact until they are rigorously verified. The term used to describe this type of claim is conjecture.

Concept

Conjecture

A conjecture is an unproven statement based on observations of a pattern. It is an educated guess that holds true for many supporting cases. A conjecture about the sum of any three consecutive numbers, for example, is shown in the table.

Sum of Any Three Consecutive Integers
Observation I Observation II Observation III
Conjecture: The sum of any three consecutive integers is three times the second number.

However, it is unknown whether any given conjecture holds true for all cases. It could be false under some circumstances and, therefore, cannot be used to support other claims. A counterexample is enough to prove that a conjecture is false.

Extra

Goldbach's Conjecture

One of the best-known conjectures is Goldbach's Conjecture, named after the German mathematician Christian Goldbach.

Goldbach's Conjecture

Every even whole number greater than is the sum of two prime numbers.

For example, the even numbers below follow the rule.
As of the conjecture has been verified by a computer for all integers less than In March it was announced that anyone who could prove Goldbach's Conjecture and whose proof was accepted by other mathematicians would be awarded a one million dollar prize. Although the prize was kept open for two years, nobody claimed it.
Discussion

Justifying a Conjecture Logically

The process of verifying that a conjecture is true is called a proof.

Concept

Proof

In mathematics, a proof is a series of logical steps of reasoning that lead to a conclusion. The reasoning should be strict enough so that the conclusion must be true if the given circumstances it uses are true.

Proof

There are several different methods that can be used to construct and visually present a mathematical proof.

Proof Methods
Visual Styles Proof Types
Paragraph Proof

Two-Column Proof

Direct Proof
The preferred method and visual style in any given case are highly dependent on the nature of the problem. The most common way of proving a certain statement is through direct proof.
Method

Direct Proof

A direct proof is a proof that uses the given information and other known facts until the statement is shown to be true. Consider the following statement.

If is an odd number, then is also an odd number.

A direct proof is dependent on what information is available, and what is the statement to be proven. The following steps summarize, in general, how to do a direct proof.
1
Understand the Given Statement
expand_more

The aim is to prove that given any odd number its square is also odd. To better understand the statement some example cases can be worked.

Is odd?
Yes
Yes
Yes
Yes

In the above table, the statement was proven to be true for just a few odd numbers, but the goal is to prove that it is true for any odd number.

2
Reach the Conclusion Using Given Information, Known Facts, and Logical Reasoning
expand_more

The given information and other known facts should be intertwined using logical reasoning several times. These can be other theorems, definitions, axioms, and so on. This step is repeated several times until a conclusion is reached.

Statement Reason
is odd Given.
Every odd number is equal to twice an integer plus
Raise the equation to the power of
Expand the square.
Factor out
is odd It is written as twice an integer plus
It is important to keep in mind that whether other theorems, definitions, or axioms are to be used or not depends on which type of statement is to be proven.
Discussion

Proven Statements

Once a conjecture is proven, it is no longer called a conjecture but a theorem.

Concept

Theorem

A theorem is a statement which is not self-evident but has been proven to be true using deductive reasoning. Many theorems come in the form of conditional statements — if-then statements that combine a and a

Pythagorean Theorem

If then

Proofs of theorems are based on definitions, axioms, corollaries, or other theorems. Therefore, theorems can be used as reasons to justify statements in the proofs of other theorems.
Discussion

Writing a Proof in a Paragraph

There are various ways of visually presenting the proof of a certain statement. One way is to write all of the statements and reasons in a single paragraph.

Method

Paragraph Proof

A paragraph proof, or informal proof, is a way of presenting a mathematical proof that consists of statements and reasons written as complete sentences in a paragraph. The reasons can be postulates, theorems, or other mathematical reasoning that the reader is assumed to be able to follow without difficulty. For example, consider the following prompt.

Let be a point on be a point on and be a point on such that and Prove that

The following steps can be used to prove this particular statement.
1
Draw a Diagram
expand_more

According to the given information, is a point on

A Point on a Line Segment

Also, it is given that is a point on and is a point on

Line segment AE with points B, C, and D dividing it into subsegments.

From the last piece of given information, is congruent to and is congruent to

Line segment AE with points B, C, and D dividing it into four subsegments, where BC is congruent to CD and AB is congruent to DE.
2
Apply the Definition of Congruence
expand_more
The given congruence statements imply that the congruent segments have equal lengths.
3
Use the Segment Addition Postulate
expand_more
The Segment Addition Postulate says that the length of a segment is the sum of the lengths of its parts.
These relationships can be visualized on the diagram.
Applying the Segment Addition Postulate
4
Use the Commutative Property of Addition
expand_more
The Commutative Property of Addition guarantees that the order of the terms in a sum can be changed.
5
Use the Substitution Property of Equality
expand_more
List all the four equations written before.
According to the Substitution Property of Equality, equal values can replace each other in equations. Substitute Equations (I) and (II) into Equation (III).
The right-hand side sum is the same as in Equation (IV). Then, substitute the left-hand side of Equation (IV) into this last equation.
6
Apply the Definition of Congruence
expand_more
If two segments have equal lengths, then the segments are congruent.
It is concluded that is congruent to
A Line Segment With Two Congruent Subsegments
Once the proof is done, it can be compactly summarized in a paragraph.
Paragraph Proof: According to the definition of congruence, and The Segment Addition Postulate states that and By the Substitution Property of Equality and the Commutative Property of Addition, it follows that Since segments with equal lengths are congruent, this completes the proof that
Example

The Detective's Hit

When the gang reached their next target, the police were already waiting for them. Finding themselves cornered, the gang split up to escape out to sea. One group flew out to sea in a helicopter. However, due to bad weather, their escape was foiled and the police captured them.
Helicopter trajectory during the storm in the sea.
According to the helicopter's radar, the angles and of the trajectory are supplementary. Write a paragraph proof that justifies this statement.

Answer

See solution.

Hint

Start by writing the given information and label some points on the trajectory to make it easier to refer to them. The triangular hatch marks indicate that the lines are parallel. When a transversal cuts two parallel lines, the alternate interior angles are congruent. If two angles form a linear pair, their measures add up to

Solution

The first step to proving a statement is identifying what information is given and what the final goal is. Labeling some points in the given diagram will make it easier to refer to them throughout the proof. For simplicity, consider only the helicopter's trajectory.

Helicopter trajectory

The triangular hatch marks indicate that and are parallel. This fact is not derived, so it is considered as given information. No more information is given, so continue by identifying what is desired to prove.

Given Prove
Angles and are supplementary

Develop logical conclusions that lead to the desired statement using the given information. For example, the alternate interior angles and are congruent because
Additionally, the angles and form a linear pair. Therefore, these two angles are supplementary.
Helicopter trajectory
From the definition of supplementary angles, their measures add up to Write an equation representing this fact.
Next, substitute for
The last equation confirms the fact that the angles and are supplementary, just as the helicopter's radar said. Finally, condense all the information written above into one paragraph.
Proof: Since the alternate interior angles and are congruent. Therefore, Also, and form a linear pair, which means they are supplementary. As such, By the Substitution Property of Equality, Consequently, the angles and are supplementary.
Discussion

Writing a Proof in a Table

Sometimes, compacting an entire proof into one paragraph results in a very long block of text that might be difficult to follow. A different way of presenting the proof is through a two-column table.

Method

Two-Column Proof

A two-column proof, or formal proof, is a compact way of showing the reasoning behind a mathematical proof. It consists of two columns, one for the statements and one for the reasons. The reasons can be postulates, theorems, or other mathematical reasoning the reader is assumed to be able to follow without difficulty. For example, consider proving the following statement.

If is the midpoint of then

Three main steps can be followed when writing a two-column proof.
1
Write the Given Information
expand_more

In the first row, write the given statement in the left-hand side column. This statement is given, not derived, so write given in the right-hand column.

Statements Reasons
is the midpoint of Given

If possible, draw a diagram that helps to derive the information that will be written in the table. This diagram will not be included in the table, though.

Segment AB with its midpoint.
2
Develop Logical Conclusions
expand_more
Starting from what is given, develop logical statements that help to prove the desired statement. Since point is the midpoint of it divides the segment into two parts of equal length.
Write this equation in the next row in the statements column. Since this equation came from the definition of a midpoint, write that in the reasons column.
Statements Reasons
is the midpoint of Given
Definition of Midpoint
3
Repeat the Step as Many Times as Necessary
expand_more

The statements written so far are not enough to reach to the desired conclusion, so continue deriving information and combining it until it points to the desired statement. Remember that postulates, theorems, or other mathematical reasoning can be used.

Statements Reasons
is the midpoint of Given
Definition of Midpoint
The Segment Addition Postulate says that the length of a segment equals the sum of the lengths of its parts. Then, the following equation can be derived.
As before, write the equation in the left-hand side and the reason in the right-hand side.
Statements Reasons
is the midpoint of Given
Definition of Midpoint
Segment Addition Postulate

Next, use the Substitution Property of Equality to substitute the equation written in the second row into the equation written in the third row.

Statements Reasons
is the midpoint of Given
Definition of Midpoint
Segment Addition Postulate
Substitution Property of Equality

The right-hand side of the last equation can be simplified by adding the two terms.

Statements Reasons
is the midpoint of Given
Definition of Midpoint
Segment Addition Postulate
Substitution Property of Equality
Simplify

Notice that the last statement is the desired one. Therefore, the proof is done!

Example

Police Chase

It is the climax of the movie. The police are chasing the rest of the gang who had escaped the last crime scene. Suddenly, the Internet connection is lost and the movie stops — right in the middle of the police chase! While waiting for the connection to be restored, Kevin tells his parents that the beams of the police car's light bar make vertical angles.
Police chase
Kevin's sister also mentions that these angles are congruent, something that impressed Kevin. Write a two-column proof to support Kevin's sister's claim.

Answer

See solution.

Hint

Start by identifying the given information and the desired statement. Label two vertical angles and consider an angle that is between them. This angle forms a linear pair with each of the labeled angles. The measures of supplementary angles add up to

Solution

Consider a pair of vertical angles, as Kevin talks about this type of angles. A diagram will be very helpful here.

Two lines forming four vertical angles
Then, Kevin's sister basically states that if two angles are vertical, then they are congruent. This statement will be proven using a two-column proof. Remember, the first step when writing a proof is to identify the given and desired statements. In this case, they can be written as follows.
It seems that just because and are vertical is not enough information to confirm that they are congruent. Some additional information is needed. Let be one of the angles between and
Two lines forming four vertical angles
From the diagram, and form a linear pair, as do and This means that the sum of the measures of each pair of angles is Write two equations representing this information.
Since both equations are equal to by the Substitution Property of Equality, they can be equated.
Finally, subtract from both sides.
The last equation is the desired statement, so the proof is done! To finish, summarize all the steps in a two-column table.
Statements Reasons
and are vertical angles Given
and form a linear pair
and form a linear pair