Lecture on Triangle Congruence Postulates

Overview of Triangle Congruence Postulates

• SSS Postulate (Side-Side-Side): If all three sides of one triangle are congruent to three sides of another triangle, the triangles are congruent.
• SAS Postulate (Side-Angle-Side): If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, the triangles are congruent.
• ASA Postulate (Angle-Side-Angle): If two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, the triangles are congruent.
• AAS Postulate (Angle-Angle-Side): If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of another, the triangles are congruent.

Examples and Applications

Example 1: SSS Postulate

• Triangles: ABC and DEF
• Congruence: AB ≅ DE, BC ≅ EF, AC ≅ DF
• Postulate Used: SSS

Example 2: ASA Postulate

• Triangles: RST and VXY
• Congruence: ∠S ≅ ∠X, RS ≅ VX, ∠R ≅ ∠V
• Postulate Used: ASA

Example 3: SAS Postulate

• Triangles: DEF and ABC
• Congruence: DE ≅ AB, DF ≅ AC, ∠D ≅ ∠A
• Postulate Used: SAS

Example 4: AAS Postulate

• Triangles: XYZ and ABC
• Congruence: XZ ≅ AC, ∠Z ≅ ∠C, ∠B ≅ ∠Y
• Postulate Used: AAS

Composite Triangles

• Example: Composite triangle with A, B, C, D
• Congruence: AD ≅ CD, AB ≅ BC, with shared side BD
• Postulate Used: SSS

Vertical Angles and Composite Figures

• Concept: Vertical angles are congruent.
• Example: Triangles involving intersecting lines with vertical angles.
• Postulate Used: ASA

Two-Column Proofs

Example Proof 1

• Given: AD ≅ CD, B is midpoint of AC
• Prove: △ABD ≅ △CBD
• Steps:
  1. AD ≅ CD (Given)
  2. B is midpoint (Given)
  3. AB ≅ BC (Definition of midpoint)
  4. BD ≅ BD (Reflexive property)
  5. △ABD ≅ △CBD (SSS)

Example Proof 2

• Given: RO ⊥ MP, MO ≅ OP
• Prove: △MRO ≅ △PRO
• Steps:
  1. MO ≅ OP (Given)
  2. RO ⊥ MP (Given)
  3. ∠MOR ≅ ∠POR (Perpendicular lines form right angles)
  4. RO ≅ RO (Reflexive property)
  5. △MRO ≅ △PRO (SAS)

Important Takeaways

• Midpoint Definition: Splits a segment into two equal parts.
• Vertical Angles: Are always congruent.
• Reflexive Property: A segment or angle is always congruent to itself.
• Congruent Supplements: Supplements of congruent angles are congruent.

Next Topics

• CPCTC (Corresponding Parts of Congruent Triangles are Congruent): Once two triangles are proven congruent, all corresponding parts are congruent.