Circle Theorems

Michelle Delfos, Oct 6, 2021

Investigate Circle Theorems

Table of Contents

  1. Labelling Geometric Features
    1. Learning about Angle Relations
    2. Lines and Angle Identification Quiz
  2. Chord Properties
    1. Chord Theorem 1 Self Explore Lesson
    2. Chord Theorem 2 Self Explore Lesson
    3. Chord Theorem 3 Self Explore Lesson
    4. Chord Theorem 4 Self Explore Lesson
    5. Practise
  3. Circle Theorems - Angles inside the circle
    1. Line from Centre to Any Chord
    2. Angle at the Centre and Angle on the Circumference
    3. Practise
    4. Chords from a Diameter to Any Point on the Circle
    5. Practise
    6. Angles on the Circumference from a Chord
    7. Practise
    8. Angles in a Cyclic Quadrilateral
    9. Practise
  4. Circle Theorems - Tangents
    1. Angle at the Tangent
    2. Tangents to a Common Point
    3. The Angle in a Cyclic Triangle and to the Tangent
    4. Practise
  5. Intersecting Chords
    1. Intersecting chords
    2. Two Sectants
    3. A sectant and a Tangent
    4. Practise

1.

Labelling Geometric Features

  • 1. Learning about Angle Relations
  • 2. Lines and Angle Identification Quiz

1.1.

Learning about Angle Relations

We have two scenarios in which we learn angle relations:[br]two lines intersecting, and two parallel lines with a transversal. (so two parallel lines with another line cutting across them)
Two Intersecting Lines - Supplementary Angles
Supplementary Angles
Supplementary Angles are any angles that add to 180.[br]If the two angles sit on the same line (such as above), we also call them "Straight Angles" because they sit on a straight line.
Two Intersecting Lines - Vertically Opposite Angles
Vertically Opposite Angles
Vertically Opposite Angles are located at the intersection of two lines, on opposite sides of the cross.[br]Unlike Straight Angles, they do not share a side.[br]They are also equivalent in size.
Alternate Angles
Alternate Angles
Alternate Angles sit on alternating sides of the Transversal. [br]They are easily spotted if they are inside the parallel lines. [br]They can also be identified by noticing the Z shape made by the side involved with angles. [br]They are of equivalent size.
Co-Interior Angles
Co-Interior Angles
Co-Interior Angles sit inside the parallel lines, and on the SAME side of the Transversal. [br]They can also be identified by noticing the C shape made by the side involved with angles. [br]They are Supplementary.
Co-Exterior Angles
Co-Exterior Angles
Co-Exterior Angles sit outside the parallel lines, and on the SAME side of the Transversal. [br]They are also Supplementary.
Corresponding Angles
Corresponding Angles
Corresponding Angles sit on the SAME side of the Transversal, but one is inside the parallel lines, and one is outside.[br]They can be identified by the F shape made by the sides involved with the angles. [br]They are also equivalent in size.
– Michelle Delfos

1.2.

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2.

Chord Properties

  • 1. Chord Theorem 1 Self Explore Lesson
  • 2. Chord Theorem 2 Self Explore Lesson
  • 3. Chord Theorem 3 Self Explore Lesson
  • 4. Chord Theorem 4 Self Explore Lesson
  • 5. Practise

2.1.

Chord Theorem 1 Self Explore Lesson

Chord Theorem 1
Chord Theorem 1
The two chords DD' and CB are set at the same length.
Chord Theorem 1
What do you notice about [math]\angle DAD'[/math] and [math]\angle CAB[/math]
– Michelle Delfos

2.2.

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2.3.

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2.4.

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2.5.

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3.

Circle Theorems - Angles inside the circle

  • 1. Line from Centre to Any Chord
  • 2. Angle at the Centre and Angle on the Circumference
  • 3. Practise
  • 4. Chords from a Diameter to Any Point on the Circle
  • 5. Practise
  • 6. Angles on the Circumference from a Chord
  • 7. Practise
  • 8. Angles in a Cyclic Quadrilateral
  • 9. Practise

3.1.

Line from Centre to Any Chord

What relationship do you notice [br]a) between [math]\angle ACD[/math] and [math]\angle ABD[/math][br]b) between [math]CD[/math] and [math]BD[/math]
– Michelle Delfos

3.2.

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3.3.

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3.4.

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3.5.

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3.6.

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3.7.

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3.8.

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3.9.

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4.

Circle Theorems - Tangents

  • 1. Angle at the Tangent
  • 2. Tangents to a Common Point
  • 3. The Angle in a Cyclic Triangle and to the Tangent
  • 4. Practise

4.1.

Angle at the Tangent

What is the angle between any radius and the tangent line?
– Michelle Delfos

4.2.

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4.3.

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4.4.

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5.

Intersecting Chords

  • 1. Intersecting chords
  • 2. Two Sectants
  • 3. A sectant and a Tangent
  • 4. Practise

5.1.

Intersecting chords

Intersecting Chords
The two chords are not related, other than intersecting.
Intersecting Chords
Multiply CF and FB.[br]Multiply DF and FE.[br][br]What do you notice for different values?
– Michelle Delfos

5.2.

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5.3.

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5.4.

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