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Understanding Congruent Figures and Proofs

Oct 31, 2024

4.1 Congruent Figures

Objective

  • Recognize congruent figures and their corresponding parts.

Key Concepts

  • Congruent Figures: Figures that have the same size and shape.

    • Examples include sliding, flipping, or turning a shape without changing its size or angles.
  • Congruent Polygons: Have congruent corresponding parts.

    • Corresponding segments are equal in length.
    • Corresponding angles are equal in measure.

Third Angle Theorem

  • If two angles of one triangle are congruent to two angles of another triangle, then the third angles are congruent.
    • Example: If angles A and D are both 30°, and angles B and E are both 50°, the third angles C and F must be congruent since triangles have 180°.

Naming Congruent Polygons

  • When naming congruent polygons, list corresponding vertices in the same order.
    • Example: If polygon ABCD is congruent to EFGH, then order matters: A corresponds to E, B to F, etc.

Corresponding Angles and Segments

  • Corresponding parts (sides and angles) are congruent.
    • Example:
      • Angles: H is congruent to L, I to M, J to N, K to O.
      • Segments: HK is congruent to OL.

Proof of Congruence

  • To prove figures are congruent, demonstrate that:
    • Three angles are congruent.
    • Three sides are congruent.
  • Proof Steps:
    1. Given: Initial congruent sides and angles provided.
    2. Identify shared or reflexive properties (e.g., shared lines or angles).
    3. Use the third angle theorem for additional angle congruence.
    4. Conclusion: If all three sides and angles are congruent, the triangles are congruent by definition.

Example Proof

  • Given: L, M congruent to L, O; M, N congruent to O, N.
  • Reflexive property: L, N is congruent to itself.
  • Third angle theorem: Additional angle congruence.
  • Conclusion: Triangle L, N, N is congruent to Triangle L, O, N.

Conclusion

  • Successfully proved congruence by demonstrating three pairs of equal sides and angles, verifying the definition of congruent triangles.