Exploring Triangle Inequalities

Objective:

Students will be able to analyse and compare angles and sides in triangles, applying the SAS Inequality Theorem and SSS Inequality Theorem.

Assessment:

Students will solve a variety of triangle inequality problems where they need to compare angles and sides in different triangles and justify their conclusions using the SAS and SSS Inequality Theorems.

Key Points:

• Triangle Inequality Relationships:

  • If one side of a triangle is longer than another side, then the angle opposite the longer side will be larger than the angle opposite the shorter side.

  • The converse is also true: If one angle in a triangle is larger than another angle in that triangle, then the side opposite the larger angle will be longer than the side opposite the smaller angle.

• SAS Inequality Theorem:

  • If two sides of a triangle are congruent to two sides of another triangle, but the included angle of one triangle has greater measure than the included angle of the other triangle, then the third side of the first triangle is longer than the third side of the second triangle.

• SSS Inequality Theorem:

  • If two sides of a triangle are congruent to two sides of another triangle, but the third side of the first triangle is longer than the third side of the second triangle, then the included angle of the first triangle's two congruent sides is greater in measure than the included angle of the second triangle's two congruent sides.

• Understanding and Application:

  • Recognizing the relationships between side lengths and angles in triangles

  • Applying the SAS Inequality Theorem to compare triangles

  • Applying the SSS Inequality Theorem to compare triangles

  • Justifying conclusions using mathematical reasoning

  • Recognizing and correcting common misconceptions about triangle inequalities

Opening:

• Engage students by presenting a visual triangle with labelled angles and sides, asking them to predict how the relationships between angles and sides might vary in different triangles.

Introduction to New Material:

• Explain the key points about triangle inequalities using real-life examples and diagrams.

• Discuss the common misconception that larger angles always correspond to longer sides in triangles.

Guided Practice:

• Provide practice problems where students compare angles and sides in triangles, gradually increasing the difficulty.

• Monitor student performance by circulating the room, asking probing questions, and providing feedback.

Independent Practice:

• Assign a worksheet with various triangle inequality problems for students to work on independently.

• Require students to justify their answers with explanations based on the SAS and SSS Inequality Theorems.

Closing:

• Have students share one interesting fact they learned about triangle inequalities with a partner.

• Summarise the key concepts of the lesson and encourage students to reflect on how they have improved their understanding.

Extension Activity:

Students who finish early can create their own triangle inequality problems for a partner to solve, challenging them to apply both the SAS and SSS Inequality Theorems.

Homework:

For homework, students are asked to find examples of triangle inequalities in real-life scenarios such as architecture, engineering, or art, and describe how the concepts of triangle inequalities are applied in those contexts.

Standards Addressed:

• CCSS.MATH.CONTENT.HSG.CO.D.12.1: Make formal geometric constructions with a variety of tools and methods.

• CCSS.MATH.CONTENT.HSG.CO.D.12.2: Derive the equations of formal geometric constructions from the geometric sketches based on given conditions.