Exploring Angles On and Inside a Circle

Objective:

Students will be able to identify and differentiate angles on and inside a circle, apply the Chord/Tangent Angle Theorem, and utilize the Intersecting Chords Angle Theorem.

Assessment:

Students will complete a worksheet where they need to identify angles on and inside a circle, solve problems using the Chord/Tangent Angle Theorem, and prove angles congruent using the Intersecting Chords Angle Theorem.

Key Points:

• Angles on a circle: When the vertex is on the edge of the circle.

• Angles inside a circle: When the vertex lies anywhere inside the circle.

• Chord/Tangent Angle Theorem: [𝛼 = 𝛽] when two chords intersect inside a circle.

• Intersecting Chords Angle Theorem: [AB * CD = BC * AD] when two chords intersect inside a circle.

Opening:

• Show a picture of a circle with various angles marked and ask students to discuss what they notice about the angles on and inside the circle

• Pose the question: "How are angles on a circle different from angles inside a circle?"

Introduction to New Material:

• Explain the definitions of angles on and inside a circle

• Demonstrate the Chord/Tangent Angle Theorem and solve a sample problem

• Common misconception: Students might confuse angles inside the circle with angles outside the circle if they are not clear on the definitions

Guided Practice:

• Provide examples of angles on and inside a circle for students to practice identifying

• Guide students through applying the Chord/Tangent Angle Theorem with scaffolded questioning

• Monitor student performance by circulating the room, offering support, and providing feedback

Independent Practice:

• Assign a set of problems requiring students to identify angles on and inside a circle, apply the Chord/Tangent Angle Theorem, and use the Intersecting Chords Angle Theorem

• Students should show all work and explanations of their solutions

Closing:

• Have students share one key concept they learned about angles on and inside a circle

• Summarize the differences between the Chord/Tangent Angle Theorem and the Intersecting Chords Angle Theorem

Extension Activity:

• Challenge students to create their own scenarios where they apply the Chord/Tangent Angle Theorem or the Intersecting Chords Angle Theorem

Homework:

• For homework, students will be asked to find real-world examples of angles on and inside a circle, take pictures, and annotate the angles identified.

Standards Addressed:

• CCSS.MATH.CONTENT.HSG-C.A.2: Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.