Exploring Triangle Congruence with ASA and AAS

Objective:

Students will be able to apply the Angle-Side-Angle (ASA) and Angle-Angle-Side (AAS) triangle congruence shortcuts to determine if two triangles are congruent.

Assessment:

Students will be given a set of triangles and will need to determine if the given information proves the two triangles congruent using either ASA or AAS.

Key Points:

• Angle-Side-Angle (ASA) Congruence Postulate: If two angles and the included side in one triangle are congruent to two angles and the included side in another triangle, then the two triangles are congruent.

  • Formula: If ∠A ≅ ∠X, ∠B ≅ ∠Y, and AB ≅ XY, then △ABC ≅ △XYZ.

• Angle-Angle-Side (AAS) Congruence Theorem: If two angles and a non-included side in one triangle are congruent to two angles and the corresponding non-included side in another triangle, then the triangles are congruent.

  • Formula: If ∠A ≅ ∠X, ∠B ≅ ∠Y, and BC ≅ YZ, then △ABC ≅ △XYZ.

• Understand the Angle-Side-Angle (ASA) Congruence Postulate and the Angle-Angle-Side (AAS) Congruence Theorem

• Identify when the given information in two triangles satisfies the conditions for ASA or AAS congruence

• Apply the congruence shortcuts to verify if two triangles are congruent

• Differentiate between ASA and AAS criteria for triangle congruence

Opening:

• Review what students know about triangle congruence

• Present a scenario where two friends are trying to prove if two triangles are congruent and ask students to predict how they will do it

Introduction to New Material:

• Explain the definitions of ASA and AAS congruence shortcuts

• Provide examples illustrating the criteria for ASA and AAS congruence

• Misconception: Emphasize that the order of the angles and sides must match in both triangles for congruence

Guided Practice:

• Show examples of triangles and guide students through determining congruence with ASA and AAS

• Scaffold questioning from simpler cases to more complex scenarios

• Monitor student performance by walking around the class and providing immediate feedback

Independent Practice:

• Students will work on a worksheet where they need to determine if the given triangles are congruent using ASA or AAS

• Encourage students to explain their reasoning for each congruence using the correct terminology

Closing:

• Have students share their findings with a partner and summarize the key differences between ASA and AAS congruence

• Review the main points of the lesson as a whole class

Extension Activity:

• Challenge early finishers to create their own pair of congruent triangles using either ASA or AAS

Homework:

• Assign students to find real-life examples where ASA or AAS congruence can be applied to prove triangles congruent

Standards Addressed:

• CCSS.MATH.CONTENT.HSG.CO.D.14a: Prove theorems about triangles

• CCSS.MATH.CONTENT.HSG.CO.D.14b: Make geometric constructions