Two-Column Proofs for Triangle Congruence

Introduction

• Focus on proving triangles congruent using two-column proofs.
• Cover five types of proofs:
  • Side-Side-Side (SSS)
  • Side-Angle-Side (SAS)
  • Angle-Side-Angle (ASA)
  • Angle-Angle-Side (AAS)
  • Hypotenuse-Leg (HL)
• Tips and techniques to simplify understanding.

Proof Example 1: Side-Angle-Side (SAS)

• Given: AB is parallel to CD, AB is congruent to CD.
• Diagram Notes: Visualize parallels with arrows and congruent sides with dashes.
• Steps:
  1. List Givens: AB parallel to CD, AB congruent to CD.
  2. Use alternate interior angles: Angle DCA is congruent to Angle BAC.
  3. AC is congruent to AC by reflexive property.
  4. Conclusion: Triangle ABC is congruent to Triangle CDA by SAS.

Proof Example 2: Side-Side-Side (SSS)

• Given: D is the midpoint of AC.
• Diagram Notes: Mark congruent sides due to midpoint.
• Steps:
  1. List Givens.
  2. Show AD congruent to CD by midpoint definition.
  3. BA congruent to BC (given).
  4. AD congruent to itself by reflexive property.
  5. Conclusion: Triangles are congruent by SSS.

Proof Example 3: Angle-Angle-Side (AAS)

• Given: Angle A congruent to Angle E, AE bisects BD.
• Diagram Notes: Mark bisected segments and angles.
• Steps:
  1. List Givens.
  2. BC congruent to DC by the definition of bisector.
  3. Vertical angles: Angle ACB congruent to Angle ECD.
  4. Conclusion: Triangles are congruent by AAS.

Proof Example 4: Angle-Side-Angle (ASA)

• Given: BD is perpendicular to AC and bisects ABC.
• Steps:
  1. List Givens.
  2. Use perpendicular definition to identify right angles.
  3. Right angles are congruent by the right angle congruence theorem.
  4. BD bisects ABC implies congruent angles ABD and CBD.
  5. Reflexive property: BD congruent to BD.
  6. Conclusion: Triangles are congruent by ASA.

Proof Example 5: Hypotenuse-Leg (HL)

• Given: ABC and DCB are right angles, AC congruent to BD.
• Diagram Notes: Consider pulling apart overlapping triangles.
• Steps:
  1. List Givens.
  2. Right angle congruence theorem applied to right angles.
  3. Reflexive property: BC congruent to BC.
  4. Conclusion: Triangles are congruent by HL theorem.

Important Notes

• Understand the importance of properly marking and identifying given elements in proofs.
• Use the appropriate theorem depending on the information available.
• Remember prohibited methods: AAA (Angle-Angle-Angle), ASS (Angle-Side-Side) are not valid for proving congruence.

Conclusion

• Practice more to get comfortable with different types of proofs.
• Utilize visual aids like diagrams to ensure clarity and accuracy.

These notes provide a structured overview of the key points learned in the lecture on proving triangle congruence using two-column proofs.