The Importance of Corollaries: Extending the Application of Theorems and Providing Additional Insights

corollary

A corollary is a statement that follows directly from a previously proven theorem or proposition

A corollary is a statement that follows directly from a previously proven theorem or proposition. It is a result or consequence of the main theorem. Corollaries are often used to provide additional insights or to extend the application of the theorem.

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

The Pythagorean Theorem states that in a right-angled 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 other two sides. This can be written as:

c^2 = a^2 + b^2

Now, a corollary of the Pythagorean Theorem could be that if the lengths of two sides of a triangle satisfy the equation a^2 + b^2 = c^2, then the triangle must be a right-angled triangle. This corollary follows from the main theorem and provides a useful application of the Pythagorean Theorem.

Corollaries are typically derived by logical reasoning and provide additional insights or consequences of a theorem. They are often used to quickly solve specific cases or to extend the application of the main theorem. Corollaries can also be used to prove other theorems or propositions by building on the previously proven results.

In summary, a corollary is a statement that follows directly from a proven theorem, providing additional insights or consequences related to the main theorem.

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