Exploring Triangle Proportionality

Objective:

Students will be able to apply the Triangle Proportionality Theorem to determine proportional segments within triangles, as well as identify and apply the Triangle Proportionality Theorem Converse.

Assessment:

Students will be given a worksheet with various triangles where they must identify proportional segments and determine if a given line is parallel to a side based on the Triangle Proportionality Theorem and its converse.

Key Points:

• Understand the Triangle Proportionality Theorem:

  • Theorem: AD/DB = AE/EC

• Understand the Triangle Proportionality Theorem Converse:

  • Converse: If AD/DB = AE/EC, then the line is parallel to the third side.

• Identify proportional segments within triangles

• Determine if a line is parallel to a side of a triangle based on proportionality

Opening:

• Engage students with a real-world scenario involving triangles and discuss the importance of proportionality within shapes.

• Pose the question: "How can we use the Triangle Proportionality Theorem to solve real-life problems involving triangles?"

Introduction to New Material:

• Present the Triangle Proportionality Theorem and its Converse through visual aids and examples.

• Discuss a common misconception: confusing proportional segments with congruent segments.

Guided Practice:

• Guide students through solving example problems step by step, gradually increasing the complexity.

• Monitor and provide feedback as students work through practice problems to ensure understanding.

Independent Practice:

• Task students to work individually or in pairs on a worksheet identifying proportional segments and determining parallel lines based on proportions within triangles.

• Provide support as needed and encourage students to explain their reasoning.

Closing:

• Have students share their findings and solutions to the practice problems with the class.

• Summarize the key concepts of the lesson and revisit the Triangle Proportionality Theorem.

Extension Activity:

• Challenge early finishers to investigate cases where the Triangle Proportionality Theorem may not apply and discuss their findings with the class.

Homework:

• Assign a set of problems for homework that reinforce the application of the Triangle Proportionality Theorem and its Converse in triangle geometry.

Standards Addressed:

• CCSS.MATH.CONTENT.HSG.SRT.B.6: Understand that by similarity, side lengths of similar polygons are proportional, and understand that by the Triangle Angle Bisector Theorem, an angle bisector in a triangle divides the opposite side into segments that are proportional to the other two sides.

• CCSS.MATH.CONTENT.HSG.SRT.B.6b: Prove theorems about triangles.