Lesson 4.1: SSS & SAS Proof Exercises

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• Lesson Description
• Learning Objectives
• Topics Covered
• Key Terms
• Lesson Examples
• Practice Problems
• Tips & Common Mistakes
• Summary
• Challenge Problems
• Unit Navigation

Lesson Description

This lesson focuses on proving triangle congruence using the Side-Side-Side (SSS) and Side-Angle-Side (SAS) theorems. Students will analyze diagrams, identify corresponding parts, and construct logical geometric proofs.

Learning Objectives

• Apply the SSS congruence theorem.
• Apply the SAS congruence theorem.
• Identify included angles in triangle diagrams.
• Write structured geometric proofs using congruence reasoning.

Topics Covered

• SSS Triangle Congruence
• SAS Triangle Congruence
• Included Angle Identification
• Corresponding Parts of Triangles
• Diagram-Based Proof Strategy

Key Terms

• SSS: If three sides of one triangle equal three sides of another, the triangles are congruent.
• SAS: If two sides and the included angle match, triangles are congruent.
• Included Angle: The angle between two known sides.
• Corresponding Parts: Matching sides and angles between triangles.

Lesson Examples

Example 1 — SSS

1. Triangle ABC has sides 5, 7, 9.
2. Triangle DEF has sides 5, 7, 9.
3. All three sides match → triangles congruent by SSS.

Example 2 — SAS

1. Two triangles share side AB.
2. Another pair of sides are equal.
3. The angle between those sides is equal.
4. Triangles congruent by SAS.

Practice Problems

1. Given three pairs of equal sides in two triangles, prove congruence.
2. Identify whether a diagram shows SAS or not. Explain why.
3. Write a two-column proof showing triangles congruent using SSS.
4. Find the included angle in a labeled triangle and determine if SAS applies.
5. Determine corresponding vertices once triangles are proven congruent.

Tips & Common Mistakes

• Always check the angle is BETWEEN the known sides for SAS.
• Do not assume triangles are congruent from two sides alone.
• Match vertices in the correct order when writing congruence statements.
• Label diagrams clearly before beginning a proof.

Summary

SSS and SAS are foundational triangle congruence rules. Mastering them allows students to formally justify triangle equality and prepares them for multi-step geometric proofs involving corresponding parts.

Challenge Problems

1. A triangle has sides 8, 10, 12. Another triangle has sides 10, 12, 8. Are they congruent? Justify.
2. Construct two triangles that share one side. Provide additional information so SAS proves them congruent.
3. Explain why SSA does NOT guarantee congruence and give a counterexample.