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Lesson Plans and Worksheets for Geometry

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More Lessons for Geometry

Common Core For Geometry

Worksheets for Geometry, Module 5, Lesson 6

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 π·πΈ and a diameter π΄π΅ are extended outside of the circle to meet at point πΆ. If πβ π·π΄πΈ = 46Β°, and πβ π·πΆπ΄ = 32Β°, find πβ π·πΈπ΄.

**Exercises**

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

- Hint: Thalesβ 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 π΄, π΅, π΅β², and πΆ are four points with π΅ and π΅β on the same side of π΄πΆβ‘ , and β π΄π΅πΆ and β π΄π΅β²πΆ are congruent, then π΄, π΅, π΅β², and πΆ 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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