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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.
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!
Here are a few recommended readings before getting started with this lesson.
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.
$n$  Number of Cubes  

Figure $1$  $0$  $1+3⋅0=1$ 
Figure $2$  $1$  $1+3⋅1=4$ 
Figure $3$  $2$  $1+3⋅2=7$ 
Figure $4$  $3$  $1+3⋅3=10$ 
Figure $121$  $120$  $1+3⋅120=361$ 
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.
Each card has a riddle written on the back that the police must solve to determine where the next robbery will take place.
Card  Green Triangles 

$1$  $1$ 
$2$  $3$ 
$3$  $6$ 
$4$  $10$ 
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  Green Triangles 

$1$  $1$ 
$2$  $3$ 
$3$  $6$ 
$4$  $10$ 
Card $2$ has $3$ green triangles, which can be written as $1+2.$ Similarly, Card $3$ has $6$ green triangles, which can be written $1+2+3.$ The same happens with Card $4.$
Card  Green Triangles 

$1$  $1=1$ 
$2$  $1+2=3$ 
$3$  $1+2+3=6$ 
$4$  $1+2+3+4=10$ 
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.
The police arrived at house $153$ 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.
Determine the answer to the thief's question using both inductive and deductive reasoning.See solution.
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 $2n+1,$ with $n$ an integer number.
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.
The thief's question involves the sum of two odd numbers. Think of two arbitrary odd numbers and calculate their sum.
Odd Numbers  Sum  

$21$  $13$  $21+(13)=34$ 
$7$  $3$  $7+3=4$ 
$5$  $5$  $5+(5)=0$ 
$1$  $5$  $1+5=6$ 
$11$  $51$  $11+51=62$ 
$33$  $45$  $33+45=78$ 
Take a look at the numbers in the righthand 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 $100%$ certainty that the conclusion is true for any two odd numbers. This is not good news to the detective.
Commutative Property of Addition
Add terms
Factor out $2$
The sum of two odd numbers is an even number.
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.
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 
$2+3+4 =9=3⋅3 $

$7+8+9 =24=3⋅8 $

$13+14+15 =42=3⋅14 $

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.
One of the bestknown conjectures is Goldbach's Conjecture, named after the German mathematician Christian Goldbach.
Goldbach's Conjecture 
Every even whole number greater than $2$ is the sum of two prime numbers. 
The process of verifying that a conjecture is true is called a 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.
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 
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 $n$ is an odd number, then $n_{2}$ is also an odd number. 
The aim is to prove that given any odd number $n,$ its square $n_{2}$ is also odd. To better understand the statement some example cases can be worked.
$n$  $n_{2}$  Is $n_{2}$ odd? 

$1$  $1$  Yes 
$3$  $9$  Yes 
$5$  $25$  Yes 
$7$  $49$  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.
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 

$n$ is odd  Given. 
$n=2k+1$  Every odd number is equal to twice an integer plus $1.$ 
$n_{2}=(2k+1)_{2}$  Raise the equation to the power of $2.$ 
$n_{2}=4k_{2}+4k+1$  Expand the square. 
$n_{2}=2(2k_{2}+2k)+1$  Factor out $2.$ 
$n_{2}$ is odd  It is written as twice an integer plus $1.$ 
Once a conjecture is proven, it is no longer called a conjecture but a theorem.
A theorem is a statement which is not selfevident but has been proven to be true using deductive reasoning. Many theorems come in the form of conditional statements — ifthen statements that combine a $hypothesis$ and a $conclusion.$
If $a triangle has a right angle,$ then $the$ $hypotenuse$ $squared$ $equals$ $the$ $sum$ $of$ $the$ $squares$ $of$ $the$ $legs.$ 
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.
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 $C$ be a point on $AE,$ $B$ be a point on $AC,$ and $D$ be a point on $CE,$ such that $AB≅DE$ and $BC≅CD.$ Prove that $AC≅CE.$ 
According to the given information, $C$ is a point on $AE.$
Also, it is given that $B$ is a point on $AC$ and $D$ is a point on $CE.$
From the last piece of given information, $AB$ is congruent to $DE$ and $BC$ is congruent to $CD.$
See solution.
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 $180_{∘}.$
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.
The triangular hatch marks indicate that $AB$ and $CE$ 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 

$AB∥CE$  Angles $α$ and $β$ are supplementary $⇕$ $α+β=180_{∘}$ 
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 twocolumn table.
A twocolumn 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 $M$ is the midpoint of $AB,$ then $AB=2AM.$ 
In the first row, write the given statement in the lefthand side column. This statement is given, not derived, so write given in the righthand column.
Statements  Reasons 
$M$ is the midpoint of $AB$  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.
Statements  Reasons 
$M$ is the midpoint of $AB$  Given 
$MB=AM$  Definition of Midpoint 
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 
$M$ is the midpoint of $AB$  Given 
$MB=AM$  Definition of Midpoint 
Statements  Reasons 
$M$ is the midpoint of $AB$  Given 
$MB=AM$  Definition of Midpoint 
$AB=AM+MB$  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 
$M$ is the midpoint of $AB$  Given 
$MB=AM$  Definition of Midpoint 
$AB=AM+MB$  Segment Addition Postulate 
$AB=AM+AM$  Substitution Property of Equality 
The righthand side of the last equation can be simplified by adding the two terms.
Statements  Reasons 
$M$ is the midpoint of $AB$  Given 
$MB=AM$  Definition of Midpoint 
$AB=AM+MB$  Segment Addition Postulate 
$AB=AM+AM$  Substitution Property of Equality 
$AB=2AM$  Simplify 
Notice that the last statement is the desired one. Therefore, the proof is done!
See solution.
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 $180_{∘}.$
Consider a pair of vertical angles, as Kevin talks about this type of angles. A diagram will be very helpful here.
Then, Kevin's sister basically states that if two angles are vertical, then they are congruent. This statement will be proven using a twocolumn 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.Statements  Reasons 
$∠1$ and $∠2$ are vertical angles  Given 
$∠1$ and $∠3$ form a linear pair $∠2$ and $∠3$ form a linear pair 
From the diagram 
$m∠1+$ 