Understanding Inscribed Angles in Circles

Objective:

Students will be able to identify and calculate inscribed angles in circles, apply the Inscribed Angle Theorem, and the Congruent Inscribed Angles Theorem.

Assessment:

Students will demonstrate mastery of the concept by correctly identifying inscribed angles and their intercepted arcs in a series of circle diagrams.

Key Points:

• Definition of Inscribed Angle: An inscribed angle is an angle with its vertex on the circle and whose sides are chords.

• Definition of Intercepted Arc: The intercepted arc is the arc that is inside the inscribed angle and whose endpoints are on the angle.

• Vertex of an Inscribed Angle: The vertex of an inscribed angle can be anywhere on the circle as long as its sides intersect the circle to form an intercepted arc.

• Inscribed Angle Theorem: This theorem states that the measure of an inscribed angle is half the measure of its intercepted arc.

• Congruent Inscribed Angles Theorem: This theorem states that if two inscribed angles intercept the same arc (or congruent arcs), then the angles are congruent.

Opening:

• Engage students by asking: "Have you ever noticed how angles inside a circle relate to the circle itself? Let's explore inscribed angles in circles today!"

Introduction to New Material:

• Define inscribed angles and intercepted arcs

• Explain the Inscribed Angle Theorem

• Anticipated misconception: Belief that the vertex of an inscribed angle must be at the center of the circle

Guided Practice:

• Provide examples of inscribed angles for students to practice identifying

• Scaffold questioning from simple to complex to ensure understanding

• Monitor student progress by circulating the room and providing guidance as needed

Independent Practice:

• Assign a worksheet with various circle diagrams requiring identification of inscribed angles and their intercepted arcs

• Students will work independently to apply the concepts taught during the lesson

Closing:

• Summarize the key concepts learned about inscribed angles in circles

• Ask students to share one new thing they learned today about inscribed angles

Extension Activity:

• For early finishers, challenge them to prove the Inscribed Angle Theorem using their understanding of inscribed angles

Homework:

• Ask students to find examples of inscribed angles in real-life situations and write a short paragraph explaining their findings

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.2: Construct the Inscribed Angle Theorem and describe its proof.