Circles

Lesson 5

Math

Unit 7

10th Grade

Lesson 5 of 14

Objective


Describe the relationship between inscribed and central angles in terms of their intercepted arc. 

Common Core Standards


Core Standards

  • G.C.A.2 — Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

Foundational Standards

  • 7.G.B.5

Criteria for Success


  1. Define inscribed angle, intercepted arc, major arc, and minor arc.
  2. Identify inscribed angles and their corresponding central angles in diagrams. See Unit 7 Glossary for a visual. 
  3. Describe that an arc can be described with an angle measure called the “arc measure.”
  4. Describe and apply the relationship between a central angle and its intercepted arc. The measure of the central angle is the same as the intercepted arc measure. See Unit 7 Glossary for a visual.
  5. Describe and apply the relationship between inscribed and central angles. The measure of the inscribed angle is always half the measure of its central angle when the vertex of the inscribed angle is in the major arc. See Unit 7 Glossary for a visual. 
  6. Given congruent central angles or inscribed angles, identify the congruent intercepted arcs. See Unit 7 Glossary for a visual.
  7. Find the missing measures of inscribed angles, central angles, and/or intercepted arcs. 

Tips for Teachers


  • In terms of pacing, this lesson might spread over two days. 
  • Ensure that students use proper notation to show the double and half relationship between arcs and angles. 
  • A common misconception students may have about this objective is the difference between “arc measure” and “arc length.” Measuring an arc like this will be new for students and is in fact the first time that they will see that an arc has a degree measure. It is important that students use the term “arc measure” in this lesson because “arc length” will be introduced in Lesson 11, which is different than arc measure.  
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Anchor Problems


Problem 1

In the diagram below, $$\angle CAB$$ is a central angle, $$\angle CDB$$ is an inscribed angle, and $$\widehat{CB}$$ is an intercepted arc.

 

  1. What do you notice about the inscribed angle and the central angle in relation to the intercepted arc? 

 

The two diagrams below also show that $$\angle CAB$$ is a central angle, $$\angle CDB$$ is an inscribed angle, and $$\widehat{CB}$$ is an intercepted arc.

  1. What is the definition of an inscribed angle and an intercepted arc? 

Guiding Questions

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Problem 2

What is the angle measure that represents $$\widehat{EB}$$?

Guiding Questions

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Problem 3

Given that $$\triangle BCD$$ is inscribed in circle $$A$$, determine the relationship between $$\angle t$$ and $$\angle y$$.

Guiding Questions

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Target Task


Given circle $$A$$ with diameters $$\overline{BC}$$ and $$\overline{DE}$$  and $$m\widehat{CD}=56^\circ$$,

  1. Name a central angle.
  2. Name an inscribed angle.
  3. Name a chord that is not a diameter.
  4. What is the measure of $$\angle CAD$$?
  5. What is the measure of $$\angle CBD$$?
  6. Name three angles of equal measure. 
  7. What is the degree measure of $$\widehat{CDB}$$?

References

EngageNY Mathematics Geometry > Module 5 > Topic B > Lesson 7Exit Ticket

Geometry > Module 5 > Topic B > Lesson 7 of the New York State Common Core Mathematics Curriculum from EngageNY and Great Minds. © 2015 Great Minds. Licensed by EngageNY of the New York State Education Department under the CC BY-NC-SA 3.0 US license. Accessed Dec. 2, 2016, 5:15 p.m..

Additional Practice


The following resources include problems and activities aligned to the objective of the lesson that can be used for additional practice or to create your own problem set.

  • Include problems that require error analysis of angle relationships, especially when showing that the central angle is half of the inscribed angle. 
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Lesson 4

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Lesson 6

Lesson Map

A7CB09C2-D12F-4F55-80DB-37298FF0A765

Topic A: Equations of Circles

Topic B: Angle and Segment Relationships in Inscribed and Circumscribed Figures

Topic C: Arc Length, Radians, and Sector Area

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