Understanding Corollaries: Exploring the Implications of Proven Math Theorems

corollary

A corollary is a statement that can be proven easily from a previously proven theorem or proposition

A corollary is a statement that can be proven easily from a previously proven theorem or proposition. It is derived directly from the main result and is seen as a immediate consequence of it. Corollaries are often used to further clarify or extend the implications of the main result, providing additional insights or applications.

When proving a corollary, you generally assume that the theorem or proposition it is based on has already been proven and you use it as a starting point. By presenting a corollary, you can show that the main result has some immediate consequences that are worth exploring as well.

To better understand the concept of a corollary, let’s consider an example:

The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

Based on this theorem, we can derive several corollaries. One such corollary is:

Corollary: In a right triangle, if the lengths of the two shorter sides are equal, then the triangle is an isosceles triangle.

Proof: Let’s assume we have a right triangle with two equal sides, let’s call them a and b, and the hypotenuse c.

According to the Pythagorean theorem, we have:

c^2 = a^2 + b^2

Since a = b, we can substitute b for a in the equation above:

c^2 = b^2 + b^2
c^2 = 2b^2

Taking the square root of both sides, we get:

c = sqrt(2b^2)
c = sqrt(2)b

Now, if c is the hypotenuse and a and b are the other two sides, we can see that c is not equal to either a or b. Therefore, the assumption that the two sides are equal leads to a contradiction. Hence, the corollary holds true.

In this example, we used the Pythagorean theorem as the starting point to prove a corollary which provides a specific condition for an isosceles triangle. The corollary is easily deduced from the main theorem and helps us understand the properties of right triangles with equal sides.

Overall, corollaries are important tools in mathematics that allow us to delve deeper into the consequences of proven theorems and expand our understanding of mathematical concepts.

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