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Proof Techniques for Congruent Triangles

Apr 25, 2025

Congruent Triangle Proofs with Quadrilaterals

Introduction

  • The video explores congruent triangle proofs involving quadrilaterals.
  • These proofs require using quadrilateral properties to achieve the desired proof outcome.

Steps in a Standard Triangle Congruency Proof

  1. Write given statements.
  2. Elaborate on given statements for keywords or symbols.
  3. Use the reflexive property for shared sides/angles.
  4. Identify vertical angles that are congruent.
  5. Prove triangle congruency.
  6. Determine if CPCTC (Corresponding Parts of Congruent Triangles are Congruent) applies.

Quadrilateral Properties

  • Parallelogram:
    • Opposite sides are congruent and parallel.
    • Opposite angles are congruent.
    • Consecutive angles are supplementary.
    • Diagonals bisect angles.
  • Rectangle:
    • Inherits properties of parallelograms.
    • Consecutive sides are perpendicular (right angles).
    • Diagonals are congruent.
  • Rhombus:
    • All sides are congruent.
    • Diagonals are perpendicular and bisect angles.
  • Square:
    • Inherits properties of parallelograms, rectangles, and rhombi.

Example Proofs

Example 1: Proving Congruency in a Rectangle

  • Given: Rectangle ABCD, diagonal AC.
  • To Prove: Triangle ADC ≅ Triangle ABC.
  • Method:
    • Used rectangle properties: opposite sides are congruent.
    • Reflexive property for shared side AC.
    • Proved triangles congruent by SSS (Side-Side-Side) congruency.

Example 2: Proving Congruency in a Parallelogram

  • Given: Parallelogram ABCD, diagonals AC and BD intersect at E.
  • To Prove: Triangle ADE ≅ Triangle CBE.
  • Method:
    • Used parallelogram properties: opposite sides and bisected diagonals.
    • Proved triangles by SSS congruency.

Example 3: A More Complex Proof with Parallelogram

  • Given: Parallelogram ABCE, angle AEF ≅ BDC.
  • To Prove: AF ≅ BD using CPCTC.
  • Method:
    • Established AE ≅ BC (opposite sides congruent).
    • Used parallel sides to find corresponding angles.
    • Proved triangles by AAS (Angle-Angle-Side) congruency.
    • Used CPCTC to conclude AF ≅ BD.

Conclusion

  • The video demonstrates how to utilize properties of quadrilaterals in triangle congruency proofs.
  • Encourages practice with multiple correct solutions possible for each proof scenario.

Full transcript