Angles in a Circle

Arthur Lee, Lyne Cyr, Oct 4, 2017

This is a series of applets for exploring properties of angles in a circle.

Table of Contents

  1. angle proportional to arc
    1. Angle at Center
    2. Angle at Center with spreadsheets
    3. Arc and Angle
    4. eyeball model
    5. Spotlight
    6. compare angles and arcs
    7. two arcs
    8. sliding secant lines
  2. angles at circumference
    1. Angle at circumference
    2. angle at center and circumference
    3. Equal angles in circle; various theorems
  3. concyclic points
    1. mark equal angles
    2. vary and mark an angle
    3. drag and trace an angle
    4. drag and trace an angle with color
    5. make 4 points concyclic
  4. tangents
    1. exploring tangent-chord angle
    2. studying tangent-chord angles
    3. tangent chord angles
  5. proof and explanation
    1. explain difference in angles with circle
    2. angle at center and circumference proof
  6. equidistance
    1. equidistant from 2 points
    2. equidistant from 3 points
    3. task: circumcenter
  7. exercise
    1. Annulus
    2. circles
    3. maximize an angle
    4. maximum angle explained
    5. task: angle and chord
    6. task: rectangle length
    7. task: chord midpoint
    8. task: make center
    9. task: make right angle
    10. task: make an angle
  8. links
    1. seeing tangent and chord from an alternate angle
    2. Circle Geometry resources in gMath

1.

angle proportional to arc

  • 1. Angle at Center
  • 2. Angle at Center with spreadsheets
  • 3. Arc and Angle
  • 4. eyeball model
  • 5. Spotlight
  • 6. compare angles and arcs
  • 7. two arcs
  • 8. sliding secant lines

1.1.

Angle at Center

Change the angle at center and the subtending arc.
Angle at Center
– Arthur Lee

1.2.

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1.3.

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1.4.

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1.5.

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1.6.

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1.7.

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1.8.

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2.

angles at circumference

  • 1. Angle at circumference
  • 2. angle at center and circumference
  • 3. Equal angles in circle; various theorems

2.1.

Angle at circumference

Compare the marked angles. When is one greater / smaller than the other?
Angle at circumference
– Arthur Lee

2.2.

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2.3.

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3.

concyclic points

  • 1. mark equal angles
  • 2. vary and mark an angle
  • 3. drag and trace an angle
  • 4. drag and trace an angle with color
  • 5. make 4 points concyclic

3.1.

mark equal angles

How can you make the angle at D the same size as the angle at B?
mark equal angles
– Arthur Lee

3.2.

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3.3.

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3.4.

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3.5.

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4.

tangents

  • 1. exploring tangent-chord angle
  • 2. studying tangent-chord angles
  • 3. tangent chord angles

4.1.

exploring tangent-chord angle

Exploring tangent-chord angle
exploring tangent-chord angle
– Arthur Lee

4.2.

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4.3.

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5.

proof and explanation

  • 1. explain difference in angles with circle
  • 2. angle at center and circumference proof

5.1.

explain difference in angles with circle

Explain how θ is greater or smaller than the angle on circumference subtended by the same arc.
explain difference in angles with circle
– Arthur Lee

5.2.

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6.

equidistance

  • 1. equidistant from 2 points
  • 2. equidistant from 3 points
  • 3. task: circumcenter

6.1.

equidistant from 2 points

equidistant from two points
equidistant from 2 points
– Arthur Lee

6.2.

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6.3.

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7.

exercise

  • 1. Annulus
  • 2. circles
  • 3. maximize an angle
  • 4. maximum angle explained
  • 5. task: angle and chord
  • 6. task: rectangle length
  • 7. task: chord midpoint
  • 8. task: make center
  • 9. task: make right angle
  • 10. task: make an angle

7.1.

Annulus

Find the area of the annulus (purple ring) based on the given the chord length of CD.
Annulus
– Arthur Lee

7.2.

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7.3.

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7.4.

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7.5.

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7.6.

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7.7.

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7.8.

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7.9.

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7.10.

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8.

links

  • 1. seeing tangent and chord from an alternate angle
  • 2. Circle Geometry resources in gMath

8.1.

seeing tangent and chord from an alternate angle

discussion about teaching tangent-chord angle theorem
Link to Google Doc [br][[url=https://docs.google.com/document/d/1yGEBZkK5cL_9S8yYMFqRuwO35XImZPqFp5unXPQlhSw/edit?usp=sharing]https://docs.google.com/document/d/1yGEBZkK5cL_9S8yYMFqRuwO35XImZPqFp5unXPQlhSw/edit?usp=sharing[/url]]
– Arthur Lee

8.2.

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