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.