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Congruence Principles in Right Triangles

May 19, 2025

Congruence in Right Triangles

Learning Goal

  • Understand congruence in right triangles.

Introduction

  • Two sides and a non-included angle of one triangle being congruent to the same in another does not guarantee congruence for all triangles.
  • Side-Side-Angle (SSA) is not a valid method for proving triangle congruence generally.
  • Special Case: In right triangles where the right angles are the non-included angles, SSA can be valid.

Right Triangle Components

  • Hypotenuse: Longest side opposite the right angle.
  • Legs: The other two sides of the triangle.

Proving Right Triangle Congruence

  • You can prove right triangles are congruent without showing all corresponding parts are congruent.
  • Hypotenuse-Leg Theorem (HL Theorem): If the hypotenuse and one leg of one right triangle are congruent to those in another, the triangles are congruent.

Proving the Hypotenuse-Leg Theorem

  1. Draw an auxiliary line to create a third triangle.
  2. Show congruence using auxiliary triangle and known congruence properties such as:
    • Transitive Property
    • Isosceles Triangle Theorem
    • Angle-Angle-Side (AAS) Theorem

Conditions for HL Theorem

  1. Two right triangles.
  2. Congruent hypotenuses.
  3. One pair of congruent legs.

Example 1: Basketball Backboard Bracket

  • Given:
    • Angle ADC and angle BDC are right angles.
    • Side AC congruent to side BC.
  • Conclusion:
    • Triangles ADC and BDC are congruent by HL Theorem.

Example 2: Proof Using HL Theorem

  • Given:
    • Segment AB is congruent to segment DE.
    • Segments AB, DE bisect at point C.
    • Angles B and E are right angles.
  • Conclusion:
    • Triangle ABC is congruent to Triangle DEC by HL Theorem.

Understanding the HL Theorem

  • It doesn't matter which leg is congruent; as long as one pair is, the theorem holds.

Practice Problem Examples

  1. Prove triangle PS is congruent to triangle RPQ:
    • Right angles at P and R.
    • Congruent hypotenuses: SP and QR.
    • Congruent legs by reflexive property.
  2. Using perpendicular bisectors and right angles to show congruence as per conditions of HL Theorem.

Lesson Check

  • Evaluate understanding by pausing and solving given problems.
  • Check correct answers and seek clarification if needed.

Self-Assessment

  • Re-evaluate your position on the learning scale after completing the lesson and exercises.

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