Complete The Missing Parts Of The Paragraph Proof.

Completing the Missing Parts of a Paragraph Proof: A Comprehensive Guide

Paragraph proofs, unlike two-column proofs, present arguments in a narrative format. This style emphasizes the logical flow of reasoning, making it easier to understand the overall structure of a mathematical proof. However, the fluidity of paragraph proofs can also make them challenging to complete when parts are missing. This guide provides a comprehensive approach to tackling such problems, equipping you with the strategies and techniques to confidently fill in the gaps.

Understanding the Structure of a Paragraph Proof

Before diving into techniques for completing missing parts, it's crucial to understand the fundamental structure of a paragraph proof. A well-structured paragraph proof typically follows this pattern:

1. Statement of the Theorem or Proposition: The proof begins by clearly stating the theorem or proposition to be proven. This sets the stage for the entire argument.

2. Assumptions and Given Information: The next step is to clearly identify any assumptions made or information given in the problem. This forms the basis upon which the argument will be built.

3. Logical Progression of Arguments: The core of the proof lies in the logical progression of arguments. Each step should follow directly from previous statements, building a chain of reasoning towards the conclusion. This often involves using definitions, axioms, postulates, previously proven theorems, or logical deductions.

4. Conclusion: The final step is to clearly state the conclusion, explicitly demonstrating that the theorem or proposition has been proven. This often involves restating the initial statement in the context of the established arguments.

Identifying Missing Parts: Common Scenarios

Missing parts in a paragraph proof can take various forms. Recognizing these common scenarios is crucial to effectively completing the proof:

1. Missing Statements:

This is the most common scenario. A crucial step or multiple steps might be absent from the proof's logical progression. Identifying these gaps requires careful analysis of the surrounding statements. Look for missing links in the chain of reasoning. Ask yourself:

• What logical step is missing to connect statement A to statement B?
• What definition, theorem, or property is needed to justify this transition?
• Are there any intermediate steps required to bridge the gap in reasoning?

2. Missing Justifications:

Another frequent issue is the omission of justifications for individual steps. Each statement within a proof must be supported by a valid reason. Missing justifications can indicate a gap in the logical flow. In such cases, you need to identify the appropriate justification for each statement, using:

• Definitions: Refer to the precise definitions of the terms used.
• Theorems and Postulates: Identify any established theorems or postulates that support the given statement.
• Logical Deductions: Determine if the statement follows logically from previous statements through deduction, such as using the transitive property or the law of syllogism.
• Properties of Equality or Inequality: Identify relevant properties like the reflexive, symmetric, transitive, addition, subtraction, multiplication, or division properties.

3. Missing Diagrams or Illustrations:

In geometric proofs, a missing diagram or illustration can significantly hinder understanding. A clear diagram can provide valuable insight into the problem, helping to visualize the relationships between elements and identify missing statements or justifications. If a diagram is missing, try to reconstruct it based on the given information.

4. Missing Conclusion:

Sometimes, the concluding statement summarizing the proven theorem is missing. In this case, you need to re-examine the entire proof's logical flow and clearly state the final conclusion. The conclusion should explicitly state what has been proven.

Strategies for Completing Missing Parts

Here's a step-by-step strategy to effectively complete the missing parts of a paragraph proof:

1. Carefully Read and Understand the Given Information: Start by thoroughly reading the provided parts of the proof. Identify the theorem to be proved, the given information, and any existing statements or justifications.

2. Identify the Gaps: Determine which parts of the proof are missing. Look for missing statements, justifications, diagrams, or a concluding statement.

3. Analyze the Logical Flow: Trace the logical flow of the existing arguments. Identify the relationships between the statements and look for missing connections or transitions.

4. Employ Relevant Definitions, Theorems, and Properties: Consult your textbook, notes, or other resources to find relevant definitions, theorems, properties, and postulates that can help fill in the missing parts.

5. Break Down Complex Steps: If a gap involves a complex step, break it down into smaller, more manageable steps. This will help to clarify the logic and make it easier to identify the necessary statements and justifications.

6. Check for Consistency: Ensure that all statements and justifications are consistent with the given information, definitions, and established theorems. Verify that each statement logically follows from the previous ones.

7. Rewrite the Complete Proof: Once you have filled in the missing parts, rewrite the entire proof in a clear, concise, and logical manner. Make sure the flow of the argument is smooth and easy to understand.

Example: Completing a Missing Part of a Geometric Proof

Let's consider an example. Suppose a portion of a geometric proof is given:

"Given: In triangle ABC, AB = AC. Angle BAC is bisected by AD, where D is a point on BC.

To Prove: Triangle ABD is congruent to triangle ACD.

Proof: Since AB = AC, triangle ABC is an isosceles triangle. AD bisects angle BAC. Therefore, angle BAD = angle CAD. … (Missing Part) … Therefore, triangle ABD is congruent to triangle ACD by SAS congruence."

Solution: The missing part involves justifying the congruence. To fill this gap, we need to establish that AD is a common side to both triangles ABD and ACD. We can add the following sentence: "Also, AD is a common side to both triangles ABD and ACD, so AD = AD by the reflexive property of equality." This completes the SAS (Side-Angle-Side) congruence criterion, completing the proof. The completed paragraph proof would then read:

"Given: In triangle ABC, AB = AC. Angle BAC is bisected by AD, where D is a point on BC.

To Prove: Triangle ABD is congruent to triangle ACD.

Proof: Since AB = AC, triangle ABC is an isosceles triangle. AD bisects angle BAC. Therefore, angle BAD = angle CAD. Also, AD is a common side to both triangles ABD and ACD, so AD = AD by the reflexive property of equality. Therefore, triangle ABD is congruent to triangle ACD by SAS congruence."

Conclusion

Completing the missing parts of a paragraph proof requires a systematic and analytical approach. By carefully analyzing the given information, identifying gaps in the logical flow, and employing relevant definitions, theorems, and properties, you can confidently construct a complete and rigorous proof. Remember to always strive for clarity, precision, and logical consistency in your arguments. Practice is key to mastering this skill, so work through various examples to build your understanding and confidence. With focused effort and consistent practice, you will become proficient in completing even the most challenging missing parts of paragraph proofs.