## Proof of the Converse of the Pythagorean Theorem

The Pythagorean Theorem states that if $ABC$ is a triangle right angled at $C$, then,

$a^2 + b^2 = c^2$.

The converse of the Pythagorean Theorem states that if

$a^2 + b^2 = c^2$ holds,

then triangle $ABC$ is a right triangle right angled at $C$.

This means that in order to prove the converse of the Pythagorean Theorem, we need to prove that $\angle C$ in the figure above is a right angle.  Now, we discuss the proof. Continue reading…

## Absolute Value Proofs: Products and Quotients

The absolute value of a number is its distance from 0. Thus, all absolute values are either positive or 0. That is, if we have a number $x$, then the absolute value of $x$ which can be written as $|x|$ is equal to

(a) $x$ if $x$ is positive

(b) $-x$ if $x$ is negative

(c) $0$ if $x$ is 0.

Also, for any real number, $x^2$ is positive or 0 (if $x = 0$).  Therefore, $\sqrt{x^2}$ is

(a) $x$ if $x$ is positive

(b) $-x$ if $x$ is negative

(c) $0$ if $x$ is 0.

If the $-x$ part is a bit confusing, consider take for example  Continue reading…

## Algebraic Proof: Product of Two Consecutive Numbers

Show that the product of $(m + 1)(m+2)$ is an even number.

This proof requires elementary knowledge in algebra and some manipulation of symbols. It is assumed that you already know or have proved that the sum of two even numbers is even, the sum of two odd integers is odd, and the sum of an odd number and an even number is odd. This proof is designed to be read by middle school and high school students, so it is written with details.

Proof 1

The numbers $m + 1$ and $m + 2$ are consecutive numbers. There are two cases possible, the smaller number is odd or even.

If the smaller number $m + 1$ is odd, then $m + 2$ is even. Now, odd multiplied by even is even.

For the second case, if the smaller number is even, then the second number is odd. Now, even multiplied by odd is always even.

Therefore, in any case the product of $(m + 1)(m + 2)$ is even.  Continue reading…