Corollary facts for kids

A corollary is like a bonus fact or a direct result that you can easily figure out once you know something else is true. It's not a completely new idea, but something that naturally follows from a main statement or rule.

What is a Corollary?

Imagine you have a big, important idea or a proven rule. A corollary is a smaller, simpler idea that becomes true automatically because the big idea is true. It's a logical step that doesn't need a lot of extra proof. Think of it as a "mini-theorem" that comes right after a "main theorem."

Corollaries in Math

In mathematics, corollaries are very common. They usually appear right after a theorem has been proven. A theorem is a statement that has been shown to be true using logical steps. Once a theorem is proven, mathematicians often notice other statements that are also true because of that theorem. These are the corollaries.

For example, if you prove that all squares have four equal sides and four right angles (a theorem), a corollary might be that a square is also a rectangle. This is true because rectangles also have four right angles, and squares fit that description perfectly. You don't need to do a whole new proof for it; it's just a direct consequence.

Why are Corollaries Important?

Corollaries are useful because they:

• Simplify understanding: They help break down complex ideas into smaller, easier-to-grasp pieces.
• Show connections: They highlight how different mathematical ideas are linked together.
• Save time: Since they follow directly from a proven theorem, you don't need to spend time proving them separately. They are "freebies" of knowledge.
• Extend knowledge: They show how a main idea can lead to other interesting facts or rules.

Examples of Corollaries

Corollaries aren't just in math! You can find similar ideas in everyday life or other subjects.

• In geometry:

* Theorem: The sum of the angles in any triangle is always 180 degrees. * Corollary: If a triangle has two angles that add up to 90 degrees, then the third angle must be 90 degrees (making it a right-angled triangle). This is a direct result of the theorem.

• In logic:

* Statement: All birds have feathers. * Corollary: If an animal does not have feathers, it cannot be a bird. This is a logical conclusion based on the first statement.

• In daily life (analogy):

* Rule: If you want to play outside, you must finish your homework first. * Corollary: If you haven't finished your homework, you cannot play outside. This is a direct consequence of the rule.

Corollaries help us build a stronger understanding of a topic by showing all the related facts that come from a main idea. They are like helpful side notes that make the main point even clearer and more useful.