New York State Common Core Math Geometry, Module 5, Lesson 6
Worksheets for Geometry, Module 5, Lesson 6
- 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
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.
Find the value 𝑥 in each figure below, and describe how you arrived at the answer.
- Hint: Thales’ theorem
- 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
- If 𝐴, 𝐵, 𝐵′, and 𝐶 are four points with 𝐵 and 𝐵’ on the same side of 𝐴𝐶⃡ , and ∠𝐴𝐵𝐶 and ∠𝐴𝐵′𝐶 are
congruent, then 𝐴, 𝐵, 𝐵′, and 𝐶 all lie on the same circle.
- 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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