Exploring Angles Outside a Circle

Objective:

Students will be able to identify and calculate angles formed by two tangents, a tangent and a secant, and two secants outside a circle.

Assessment:

Students will be given a worksheet with various diagrams of circles and angles outside the circle. They must identify the type of angle (formed by two tangents, a tangent and a secant, or two secants) and calculate the angle measure.

Key Points:

• Definition: An angle is outside a circle if its vertex is outside the circle and its sides are tangents or secants.

• Possibilities: The possibilities for angles outside a circle include an angle formed by two tangents, an angle formed by a tangent and a secant, and an angle formed by two secants.

• Outside Angle Theorem: The Outside Angle Theorem states that when two secants, a secant and a tangent, or two tangents intersect outside a circle, the measure of the angle formed is equal to half the difference of the measures of the intercepted arcs.

Opening:

• Engage students with a real-life scenario involving angles outside a circle (e.g., a Ferris wheel or a bicycle wheel)

• Pose the question: "Have you ever noticed angles formed outside circles in everyday life? How are they different from angles inside a circle?"

Introduction to New Material:

• Define what it means for an angle to be outside a circle

• Present examples of angles formed by two tangents, a tangent and a secant, and two secants outside a circle

• Address the common misconception that angles outside a circle are always obtuse

Guided Practice:

• Provide guided examples for students to practice identifying and calculating angles outside a circle

• Scaffold questioning from easier examples to more complex scenarios

• Monitor student performance through questioning and one-on-one support

Independent Practice:

• Assign students a worksheet with various angles outside circles to identify and calculate

• Encourage students to show their work and explain their reasoning in detail

• Monitor students as they work independently, providing assistance as needed

Closing:

• Conduct a quick group discussion to review the key concepts learned during the lesson

• Summarize the different types of angles formed outside a circle and their properties

Extension Activity:

• Challenge early finishers to construct their own diagrams of circles with angles outside the circle and exchange them with a partner to solve

• Encourage students to create a real-world scenario involving angles outside a circle that they can share with the class

Homework:

• Suggest homework activity: Students are to find three real-life examples of angles formed outside circles and identify which type of angle each one is (formed by two tangents, a tangent and a secant, or two secants).

Standards Addressed:

• CCSS.MATH.CONTENT.HSG-C.A.2: Identify and describe relationships among inscribed angles, radii, and chords.

• CCSS.MATH.CONTENT.HSG-C.A.3: Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.