Unknown Angle Problems with Inscribed Angles in Circles

New York State Common Core Math Geometry, Module 5, Lesson 6

Worksheets for Geometry

Student Outcomes

• Use the inscribed angle theorem to find the measures of unknown angles.
• Prove relationships between inscribed angles and central angles.

Unknown Angle Problems with Inscribed Angles in Circles

Classwork

Opening Exercise

In a circle, a chord 𝐷𝐸 and a diameter 𝐴𝐵 are extended outside of the circle to meet at point 𝐶. If 𝑚∠𝐷𝐴𝐸 = 46°, and 𝑚∠𝐷𝐶𝐴 = 32°, find 𝑚∠𝐷𝐸𝐴.

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Exercises

Find the value 𝑥 in each figure below, and describe how you arrived at the answer.

1. Hint: Thales’ theorem

Lesson Summary

Theorems:

• THE INSCRIBED ANGLE THEOREM: The measure of a central angle is twice the measure of any inscribed angle that intercepts the same arc as the central angle.
• CONSEQUENCE OF INSCRIBED ANGLE THEOREM: Inscribed angles that intercept the same arc are equal in measure.
• If 𝐴, 𝐵, 𝐵′, and 𝐶 are four points with 𝐵 and 𝐵’ on the same side of 𝐴𝐶 , and ∠𝐴𝐵𝐶 and ∠𝐴𝐵′𝐶 are congruent, then 𝐴, 𝐵, 𝐵′, and 𝐶 all lie on the same circle.

Relevant Vocabulary

• CENTRAL ANGLE: A central angle of a circle is an angle whose vertex is the center of a circle.
• 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.
• INTERCEPTED ARC: 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. An angle inscribed in a circle intercepts exactly one arc, in particular, the arc intercepted by a right angle is the semicircle in the interior of the angle.

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