Exploring Angle Bisectors in Triangles

Objective:

Students will be able to demonstrate an understanding of angle bisectors in triangles by applying the Angle Bisector Theorem to solve problems and prove properties related to angle bisectors.

Assessment:

Students will be assessed through a task where they identify angle bisectors in given triangles, calculate missing angles using the Angle Bisector Theorem, and provide written explanations for how the theorem applies to specific scenarios.

Key Points:

• Definition of an angle bisector in a triangle: An angle bisector in a triangle is a line segment that divides an angle into two equal angles.

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

• Angle Bisector Theorem Converse: If a point is equidistant from the sides of an angle, then it lies on the angle bisector.

Formulas and Definitions:

• Angle Bisector Theorem: If AD is the angle bisector in triangle ABC, then AB/AC = BD/DC.

• Angle Bisector Theorem Converse: If a point P is equidistant from the sides AB and AC of angle A in triangle ABC, then AP is the angle bisector of angle A.

Visual Representation:

• Include diagrams illustrating the Angle Bisector Theorem in action, showing how angle bisectors divide angles and sides proportionally.

• Provide visual examples of the Angle Bisector Theorem Converse, demonstrating points equidistant from the sides of an angle lying on the angle bisector.

Opening:

• Engage students by asking them to think about a time they had to divide something equally. How did they ensure it was divided exactly in half? Discuss with a partner and share with the class.

Introduction to New Material:

• Explain the concept of angle bisectors and how they cut angles in half.

• Demonstrate the Angle Bisector Theorem using diagrams and interactive examples.

• Common misconception: Students may confuse angle bisectors with perpendicular bisectors, so clarify the distinction clearly.

Guided Practice:

• Provide triangles for students to identify and draw angle bisectors.

• Walk students through solving problems using the Angle Bisector Theorem step by step.

• Ask questions to scaffold learning from basic identification to more complex problem-solving.

• Monitor student progress by circulating the room, providing guidance, and checking for understanding.

Independent Practice:

• Assign a worksheet where students identify angle bisectors, calculate missing angles, and explain their reasoning.

• Encourage students to work independently but collaborate with peers if needed.

• Provide additional problems of varying difficulty to challenge different levels of understanding.

Closing:

• Summarise the key points of the lesson by asking students to explain the Angle Bisector Theorem in their own words.

• Have students share their responses with a partner before discussing as a class.

Extension Activity:

• Challenge early finishers to explore the converse of the Angle Bisector Theorem: If a point is equidistant from the sides of an angle, is it on the angle bisector?

Homework:

• Homework Activity: Provide a set of triangles and ask students to identify angle bisectors in each triangle, measure the angles involved, and write a short paragraph explaining how the Angle Bisector Theorem is applied.

Standards Addressed:

• CCSS.MATH.CONTENT.HSG.CO.C.8a: Apply the Angle Bisector Theorem to solve problems

• CCSS.MATH.CONTENT.HSG.CO.C.8b: Prove theorems about triangles