Lecture Notes: Triangle Congruence by ASA and AAS

Learning Goals

• Understand how to prove triangle congruence using different postulates and theorems.
• Evaluate your understanding before and after the lesson.

Review of Previous Lessons

• Side-Angle-Side (SAS) Postulate: Triangles are congruent if two pairs of sides and the included angle are congruent.

Current Lesson: Angle-Side-Angle (ASA) Postulate

• Definition: Two triangles are congruent if two angles and the included side of one triangle are congruent to two angles and the included side of another triangle.
  • Example: If angle A = angle D, side AC = side DF, and angle C = angle F, then triangle ABC is congruent to triangle DEF by ASA.
  • Key Requirement: The side must be included between the two angles.
  • Example: Triangle SUV and Triangle ONE are congruent because the included side is between the two angles.

Proving Congruence with ASA

• Proof Example 1:

  • Given: Side AB = side DE, angle A = angle D, angles B and E are right angles.
  • All right angles are congruent: angle B = angle E.
  • Conclusion: Triangle ABC is congruent to Triangle DEF by ASA.

• Proof Example 2:

  • Given: Angle CA = angle DAE, side BA = side EA, angles B and E are right angles.
  • Conclusion: Triangle ABC is congruent to Triangle AED by ASA.

Angle-Angle-Side (AAS) Theorem

• Definition: If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of another triangle, then the triangles are congruent.
  • Example: Angle A = angle D, angle B = angle E, side AC = side DF.

Proving Congruence with AAS

• Proof Example 3:

  • Given: Angle M = angle K, sides WN and RK are parallel.
  • Alternate Interior Angles Theorem: angle MWR = angle KRW.
  • Reflexive Property: side WR = side RW.
  • Conclusion: Triangle WMR is congruent to Triangle RKW by AAS.

• Alternate Proof Using ASA:

  • Given: Same initial conditions as above.
  • Conclusion: Triangle WMR is congruent to Triangle RKW by ASA.

Determining Congruence

• Example 4:

  • Given two triangles with two angles and a non-included side.
  • Corresponding sides do not match; therefore, triangles are not congruent.

• Exercise: Evaluate triangle congruence using given conditions and determine the correct postulate/theorem to use.

Lesson Wrap-up

• Review your understanding against the learning goals.
• Check answers and review any uncertainties in the next class.
• Attempt the challenge problem if confident.

Self-Assessment

• Re-evaluate your position on the learning scale.
• Prepare any questions for the next class session.