Main Question

Given two triangles ABC and DEF where AB = DE, AC = DF, and ∠A = ∠D, prove that the triangles are congruent.

More importantly, can you explain why we need exactly these three pieces of information and not others?

Key Concepts to Explore

• The difference between the four congruence theorems (SSS, SAS, ASA, AAS)
• Why AAA doesn’t guarantee congruence
• The logic behind geometric proofs
• How congruence relates to transformations (rotations, reflections, translations)

Expected Follow-up Questions

High-quality student engagement should include questions like:

• “Why doesn’t SSA work as a congruence theorem?”
• “How do I know which theorem to use?”
• “What’s the difference between congruent and similar?”
• “Why do we need formal proofs in geometry?”
• “Can I prove congruence with different combinations of sides and angles?”
• “How do I write a clear, logical proof?”

Critical Thinking Indicators

Students demonstrate understanding when they:

• Question why certain combinations work while others don’t
• Want to understand the logic behind proof structure
• Explore counterexamples (like why AAA fails)
• Connect congruence to real-world applications
• Ask about the relationship between congruence and similarity
• Investigate what happens with different given information
• Show curiosity about the historical development of geometric proofs

Additional Exploration

• Construct triangles with different given information to test congruence
• Explore how rigid transformations relate to congruence
• Investigate special cases (right triangles, isosceles triangles)
• Connect to coordinate geometry and distance formulas

Assessment Criteria

Strong responses should include:

• Correct identification of the SAS congruence theorem
• Clear, step-by-step proof structure
• Understanding of why the given information is sufficient
• Ability to distinguish this case from others (like SSA)
• Connection between algebraic reasoning and geometric visualization