HOW DO I SUPPLY THE AXIOM OR THEOREM THAT JUSTIFIES TO A GIVEN STATEMENT
You have to know what axiom or theorem to fill in. It depends upon the statement you are trying to prove. The proof of theorems in geometry, or in math, depends upon knowledge of previous axioms and theorems. You just have to learn them.
To supply an axiom or theorem that justifies a given statement, you need to follow a logical and deductive reasoning process. Here's a general step-by-step approach to help you accomplish this:
1. Understand the statement: Clearly comprehend what the given statement is asserting. Break it down into smaller, more manageable components if necessary.
2. Identify relevant definitions: Examine the terms and concepts used in the statement. Look for definitions that provide meaning to these terms. A definition could be an axiom itself or derived from previous axioms or theorems.
3. Examine available axioms: Review your list of axioms, which are foundational statements that are considered self-evident truths. Identify any axioms that directly apply to the statement at hand. If an axiom is relevant, you can directly cite it as the justification for the statement.
4. Apply relevant theorems: Look for theorems that have been proven or derived from previously established statements or axioms. Check if any of these theorems can be used to prove or justify the given statement. If you find a relevant theorem, cite it as the justification for the statement.
5. Prove from scratch: If you cannot find a specific axiom or theorem to justify the given statement, you may need to build a proof from scratch. In this case, start with other proven statements or previously established theorems relevant to the given statement, and apply logical deductions or reasoning steps to derive the desired statement.
6. Provide a clear explanation: Once you have determined the appropriate axiom or theorem to justify the statement, clearly state the justification. Explain how the chosen axiom or theorem directly relates to the given statement and why it is applicable. Use logical reasoning and clear language to make your explanation coherent.
Remember that the approach may vary depending on the specific field of study or the nature of the statement. Always consult relevant textbooks, references, or expert opinions to ensure you are using accurate axioms or theorems.