Introduction to Theorems

In geometry, we often come across statements that we know are true because we can prove them. These statements are called theorems. Understanding what a theorem is and how it works is important for solving maths problems, especially in geometry.

To make sense of theorems, we also need to understand a few other terms that are connected to them. Let's go through each one step by step.

1. Axiom

• What It Is: An $axiom$ is like a basic rule that everyone agrees on without needing to prove it. It's something we just accept as true.

• Why It Matters: Axioms are like the starting points in maths. We use them to build more complicated ideas.

2. Theorem

• What It Is: A $theorem$ is a statement that we can prove to be true. We don't just accept it—we actually show, step by step, why it's always true.

• Why It Matters: Theorems are the "rules" of geometry that we can always count on because we've proved them.

3. Proof

• What It Is: A $proof$ is like a set of instructions that shows why a theorem is true. It's a series of steps that lead you from what you know (like axioms or other theorems) to the new statement you're trying to prove.

• Why It Matters: Proofs help us understand why theorems work, so we're not just memorizing facts—we're seeing the logic behind them.

4. Corollary

• What It Is: A $corollary$ is like a "bonus" fact that comes from a theorem. Once you've proven a theorem, a corollary is something extra you can figure out without much more work.

• Why It Matters: Corollaries help you get more information from a theorem, making problem-solving quicker and easier.

5. Converse

• What It Is: The $converse$ of a statement is what you get when you flip the statement around. Instead of saying "If $P$, then $Q,S$" the converse would be "If $Q$, then $P$."

• Why It Matters: Sometimes, the converse of a theorem is true, and sometimes it's not. It's important to check whether both the original statement and its converse are correct.

6. Implies

• What It Is: "$Implies$" is a word we use to show that one statement leads to another. If you know that one thing is true, it might "imply" that something else is true too.

• Symbol: The symbol for "$implies$" is "$=>$". This means "If this is true, then that must also be true."

Why This Matters

Understanding these terms is like learning the rules of the game in maths. Once you know what a theorem is and how it works, you can use these ideas to solve all sorts of problems. It's like building a strong foundation that you can rely on as you tackle more complex maths.

---