The Angle Measure of an Arc

New York State Common Core Math Geometry, Module 5, Lesson 7

Worksheets for Geometry

Student Outcomes

• Define the angle measure of arcs, and understand that arcs of equal angle measure are similar.
• Restate and understand the inscribed angle theorem in terms of arcs: The measure of an inscribed angle is half the angle measure of its intercepted arc.
• Explain and understand that all circles are similar.

The Angle Measure of an Arc

Classwork

Opening Exercise

If the measure of ∠𝐺𝐵𝐹 is 17°, name three other angles that have the same measure and explain why.
What is the measure of ∠𝐺𝐴𝐹? Explain.
Can you find the measure of ∠𝐵𝐴𝐷? Explain.

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Example

What if we started with an angle inscribed in the minor arc between 𝐴 and 𝐶?

Exercises

1. In circle 𝐴, 𝑚𝐵𝐶 : 𝑚𝐶𝐸 : 𝑚𝐸𝐷 : 𝑚𝐷𝐵 = 1: 2: 3: 4. Find the following angles of measure.
   a. 𝑚∠𝐵𝐴𝐶
   b. 𝑚∠𝐷𝐴𝐸
   c. 𝑚𝐷𝐵
   d. 𝑚𝐶𝐸𝐷
2. In circle 𝐵, 𝐴𝐵 = 𝐶𝐷. Find the following angles of measure.
   a. 𝑚𝐶𝐷
   b. 𝑚𝐶𝐴𝐷
   c. 𝑚𝐴𝐶𝐷
3. In circle 𝐴, 𝐵𝐶 is a diameter and 𝑚∠𝐷𝐴𝐶 = 100°. If 𝑚𝐸𝐶 = 2𝑚𝐵𝐷 , find the following angles of measure.
   a. 𝑚∠𝐵𝐴𝐸
   b. 𝑚𝐸𝐶
   c. 𝑚𝐷𝐸𝐶
4. Given circle 𝐴 with 𝑚∠𝐶𝐴𝐷 = 37°, find the following angles of measure.
   a. 𝑚𝐶𝐵𝐷
   b. 𝑚∠𝐶𝐵𝐷
   c. 𝑚∠𝐶𝐸𝐷

Lesson Summary

Theorems:

• INSCRIBED ANGLE THEOREM: The measure of an inscribed angle is half the measure of its intercepted arc.
• Two arcs (of possibly different circles) are similar if they have the same angle measure. Two arcs in the same or congruent circles are congruent if they have the same angle measure.
• All circles are similar

Relevant Vocabulary

• ARC: An arc is a portion of the circumference of a circle.
• MINOR AND MAJOR ARC: Let 𝐶 be a circle with center 𝑂, and let 𝐴 and 𝐵 be different points that lie on 𝐶 but are not the endpoints of the same diameter. The minor arc is the set containing 𝐴, 𝐵, and all points of 𝐶 that are in the interior of ∠𝐴𝑂𝐵. The major arc is the set containing 𝐴, 𝐵, and all points of 𝐶 that lie in the exterior of ∠𝐴𝑂𝐵.
• SEMICIRCLE: In a circle, let 𝐴 and 𝐵 be the endpoints of a diameter. A semicircle is the set containing 𝐴, 𝐵, and all points of the circle that lie in a given half-plane of the line determined by the diameter.
• INSCRIBED ANGLE: An inscribed angle is an angle whose vertex is on a circle and each side of the angle intersects the circle in another point.
• CENTRAL ANGLE: A central angle of a circle is an angle whose vertex is the center of a circle.
• INTERCEPTED ARC OF AN ANGLE: An angle intercepts an arc if the endpoints of the arc lie on the angle, all other points of the arc are in the interior of the angle, and each side of the angle contains an endpoint of the arc.

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