Geometric Proof - Geometry in Two Dimensions (Pre-Algebra)

	$n$	Number of Cubes
Figure $1$	$0$	$1 + 3 \cdot 0 = 1$
Figure $2$	$1$	$1 + 3 \cdot 1 = 4$
Figure $3$	$2$	$1 + 3 \cdot 2 = 7$
Figure $4$	$3$	$1 + 3 \cdot 3 = 10$
Figure $121$	$120$	$1 + 3 \cdot 120 = 361$

n

Number of Cubes

Figure

1

0

1 + 3 \cdot 0 = 1

Figure

2

1

1 + 3 \cdot 1 = 4

Figure

3

2

1 + 3 \cdot 2 = 7

Figure

4

3

1 + 3 \cdot 3 = 10

Figure

121

120

1 + 3 \cdot 120 = 361

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

Card

Green Triangles

1

1

2

3

3

6

4

10

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

Card

Green Triangles

1

1

2

3

3

6

4

10

Card	Green Triangles
$1$	$1 = 1$
$2$	$1 + 2 = 3$
$3$	$1 + 2 + 3 = 6$
$4$	$1 + 2 + 3 + 4 = 10$

Card

Green Triangles

1

1 = 1

2

1 + 2 = 3

3

1 + 2 + 3 = 6

4

1 + 2 + 3 + 4 = 10

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
$- 21$	$- 13$	$- 21 + (- 13) = - 34$
$- 7$	$3$	$- 7 + 3 = 4$
$5$	$- 5$	$5 + (- 5) = 0$
$1$	$5$	$1 + 5 = 6$
$11$	$51$	$11 + 51 = 62$
$33$	$45$	$33 + 45 = 78$

Take a look at the numbers in the 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 $100 %$ 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

2 n + 1,

where

n

is an integer number. Let

A

and

B

be two different odd numbers.

A B = 2 n + 1 = 2 m + 1

Since the purpose is to study the sum of two odd numbers, add

A

and

B

and simplify as much as possible.

A + B = 2 n + 1 + 2 m + 1

CommutativePropAdd

Commutative Property of Addition

A + B = 2 n + 2 m + 1 + 1

AddTerms

Add terms

A + B = 2 n + 2 m + 2

FactorOut

Factor out $2$

A + B = 2 (n + m + 1)

Notice that the sum of

A

and

B

is equal to

2

times

(n + m + 1) .

Since

n

and

m

are integers, the sum inside the parentheses is an integer. Then, the sum of the two odd numbers has the form

2 K,

where

K = n + m + 1 .

A + B = 2 K

This form matches the general form of even numbers. Therefore,

A + B

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.

Deductive Reasoning ↘ Inductive Reasoning ↙ The sum of two odd numbers is an even number .

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.

Sum of Any Three Consecutive Integers
Observation I	Observation II	Observation III
$2 + 3 + 4 = 9 = 3 \cdot 3$	$7 + 8 + 9 = 24 = 3 \cdot 8$	$13 + 14 + 15 = 42 = 3 \cdot 14$
Conjecture: The sum of any three consecutive integers is three times the second number.

Sum of Any Three Consecutive Integers

Observation I

Observation II

Observation III

2 + 3 + 4 = 9 = 3 \cdot 3

7 + 8 + 9 = 24 = 3 \cdot 8

13 + 14 + 15 = 42 = 3 \cdot 14

Conjecture: The sum of any three consecutive integers is three times the second number.

Goldbach's Conjecture
Every even whole number greater than $2$ is the sum of two prime numbers.

Proof Methods
Visual Styles	Proof Types
Paragraph Proof Two-Column Proof	Direct Proof

Proof Methods

Visual Styles

Proof Types

Paragraph Proof

Two-Column Proof

Direct Proof

If $n$ is an odd number, then $n^{2}$ is also an odd number.

$n$	$n^{2}$	Is $n^{2}$ odd?
$1$	$1$	Yes
$3$	$9$	Yes
$5$	$25$	Yes
$7$	$49$	Yes

n

n^{2}

n^{2}

odd?

1

1

Yes

3

9

Yes

5

25

Yes

7

49

Yes

Statement	Reason
$n$ is odd	Given.
$n = 2 k + 1$	Every odd number is equal to twice an integer plus $1 .$
$n^{2} = (2 k + 1)^{2}$	Raise the equation to the power of $2 .$
$n^{2} = 4 k^{2} + 4 k + 1$	Expand the square.
$n^{2} = 2 (2 k^{2} + 2 k) + 1$	Factor out $2 .$
$n^{2}$ is odd	It is written as twice an integer plus $1 .$

Statement

Reason

n

is odd

Given.

n = 2 k + 1

Every odd number is equal to twice an integer plus

1 .

n^{2} = (2 k + 1)^{2}

Raise the equation to the power of

2 .

n^{2} = 4 k^{2} + 4 k + 1

Expand the square.

n^{2} = 2 (2 k^{2} + 2 k) + 1

Factor out

2 .

n^{2}

is odd

It is written as twice an integer plus

1 .

Pythagorean Theorem
If $a triangle has a right angle,$ then $the$ $hypotenuse$ $squared$ $equals$ $the$ $sum$ $of$ $the$ $squares$ $of$ $the$ $legs .$

Let $C$ be a point on $\overline{A E},$ $B$ be a point on $\overline{A C},$ and $D$ be a point on $\overline{C E},$ such that $\overline{A B} ≅ \overline{D E}$ and $\overline{B C} ≅ \overline{C D} .$ Prove that $\overline{A C} ≅ \overline{C E} .$

Given	Prove
$\overline{A B} ∥ \overline{C E}$	Angles $α$ and $β$ are supplementary $⇕$ $α + β = 18 0^{\circ}$

Given

Prove

\overline{A B} ∥ \overline{C E}

Angles

α

and

β

are supplementary

⇕

α + β = 18 0^{\circ}

If $M$ is the midpoint of $\overline{A B},$ then $A B = 2 A M .$

$2$ is an irrational number.

	Description
Indirect Proof or Proof by Contradiction	A claim is proven by showing how the opposite conclusion of the claim creates a contradiction.

Description

Indirect Proof or Proof by Contradiction

A claim is proven by showing how the opposite conclusion of the claim creates a contradiction.

	Description
Proof by Contrapositive	A claim is proven by stating and proving the contrapositive of the claim.
Proof by Mathematical Induction	One special case is shown to be true. Next, it is shown that if the statement is true for any special case, then it is also true for some other special case.
Coordinate Proof	It is used to prove geometric statements by placing geometric figures in a coordinate plane and assigning variables to the coordinates of points.

Description

Proof by Contrapositive

A claim is proven by stating and proving the contrapositive of the claim.

Proof by Mathematical Induction

One special case is shown to be true. Next, it is shown that if the statement is true for any special case, then it is also true for some other special case.

Coordinate Proof

It is used to prove geometric statements by placing geometric figures in a coordinate plane and assigning variables to the coordinates of points.

Statements	Reasons
$M$ is the midpoint of $\overline{A B}$	Given

Statements	Reasons
$M$ is the midpoint of $\overline{A B}$	Given
$M B = A M$	Definition of Midpoint

Statements	Reasons
$M$ is the midpoint of $\overline{A B}$	Given
$M B = A M$	Definition of Midpoint

Statements	Reasons
$M$ is the midpoint of $\overline{A B}$	Given
$M B = A M$	Definition of Midpoint
$A B = A M + M B$	Segment Addition Postulate

Statements	Reasons
$M$ is the midpoint of $\overline{A B}$	Given
$M B = A M$	Definition of Midpoint
$A B = A M + M B$	Segment Addition Postulate
$A B = A M + A M$	Substitution Property of Equality

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Catch-Up and Review

Which Is More Reliable?

Looking for Patterns

Inductive Reasoning

The Thief's Pattern

Hint

Solution

Logical Reasoning

Deductive Reasoning

An Odd Riddle

Answer

Hint

Solution

Using Inductive Reasoning

Using Deductive Reasoning

Conclusion

Unproven Statements

Conjecture

Extra

Justifying a Conjecture Logically

Proof

Direct Proof

Proven Statements

Theorem

Writing a Proof in a Paragraph

Paragraph Proof

The Detective's Hit

Answer

Hint

Solution

Writing a Proof in a Table

Two-Column Proof

Police Chase

Answer

Hint

Solution

Other Types of Proofs

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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
$m ∠ 1 + m ∠ 3$ $= 18 0^{\circ}$ $m ∠ 2 + m ∠ 3$ $= 18 0^{\circ}$	Definition of linear pair
$m ∠ 1 + m ∠ 3 =$ $m ∠ 2 + m ∠ 3$	Substitution Property of Equality
$m ∠ 1 = m ∠ 2$	Subtraction Property of Equality