Exploring Proportions with Angle Bisectors

Objective:

Students will be able to apply the Angle Bisector Theorem to identify and solve proportions within triangles.

Assessment:

Students will demonstrate mastery of the objective by completing a worksheet containing various triangles where they will identify angle bisectors, determine proportional segments, and solve for missing side lengths using the Angle Bisector Theorem.

Key Points:

• Understand the Angle Bisector Theorem: If a ray bisects an angle of a triangle, then it divides the opposite side into segments that are proportional to the lengths of the other two sides.

• Identify angle bisectors within triangles

• Solve for proportional segments within triangles

• Apply the Angle Bisector Theorem to find missing side lengths in triangles

Definitions and Formulas:

• Angle Bisector Theorem: If a ray bisects an angle of a triangle, then it divides the opposite side into segments that are proportional to the lengths of the other two sides.

Opening:

• Engage students by asking: "How can we use angle bisectors to find proportional segments within a triangle?"

Introduction to New Material:

• Explain the Angle Bisector Theorem and its application

• Demonstrate how to identify and work with angle bisectors in triangles

• Common misconception to anticipate: Students may confuse angle bisectors with medians or altitudes of triangles

Guided Practice:

• Provide examples of triangles where students can identify angle bisectors

• Scaffold questioning from simple to complex to deepen understanding

• Monitor student performance by circulating the classroom and providing guidance as needed

Independent Practice:

• Assign a worksheet with various triangles for students to practice identifying angle bisectors and solving for proportional segments

• Behavioral expectations: Work quietly, show all work, and ask for help when needed

Closing:

• Summarise key concepts learned by discussing how the Angle Bisector Theorem helps us understand proportions within triangles

Extension Activity:

• For early finishers, create a task where they can explore real-world examples of angle bisectors in architecture or design

Homework:

• Provide a homework activity where students need to find and sketch examples of angle bisectors in everyday objects or situations

Standards Addressed:

• CCSS.MATH.CONTENT.HSG.CO.C.9: Prove theorems about triangles, including the Angle Bisector Theorem and its converse.