Mathematics - Senior Secondary 2 - Circle theorem (Angle theorems)

Circle theorem (Angle theorems)

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TERM: 2ND TERM

WEEK: 5

Class: Senior Secondary School 2

Age: 16 years

Duration: 40 minutes of 5 periods

Subject: Mathematics

Topic: Circle Theorem (Angle Theorems)

Focus: Understanding and applying key theorems related to angles in circles, including:

  1. The angle at the center is twice the angle at the circumference.
  2. Angles in the same segment are equal.
  3. The angle in a semicircle is 90°.
  4. Opposite angles of a cyclic quadrilateral are supplementary.
  5. Tangent to a circle (radius is perpendicular to the tangent).

 

SPECIFIC OBJECTIVES:

By the end of the lesson, students should be able to:

  1. Understand and prove the theorem that the angle at the center is twice the angle at the circumference.
  2. Identify and explain that angles in the same segment of a circle are equal.
  3. Demonstrate and prove that the angle in a semicircle is 90°.
  4. Identify and prove that opposite angles of a cyclic quadrilateral are supplementary.
  5. Explain and prove that the radius of a circle is perpendicular to the tangent.

 

INSTRUCTIONAL TECHNIQUES:

  • Guided demonstration (with the use of models and diagrams).
  • Question and answer (to assess students’ understanding and probe deeper).
  • Practical application (by drawing diagrams and performing proofs).
  • Discussion and group work (for peer learning and sharing ideas).
  • Real-life applications (to link theory to practical scenarios).

 

INSTRUCTIONAL MATERIALS:

  • Cardboard (for constructing models).
  • Protractors and rulers (for measuring angles).
  • Diagrams illustrating different circle theorems.
  • Whiteboard and markers for explanations and note-taking.

 

PERIOD 1 & 2: Angle at the Centre and Angle at the Circumference

PRESENTATION:

Step 1 - Introduction to the Angle at the Centre Theorem

  • Teacher explains that the angle at the center of a circle is twice the angle at the circumference subtended by the same chord.
  • Teacher draws a circle with a chord and labels the center and the circumference.
  • Example: If ∠AOC is 40°, then ∠ABC (on the circumference) is 20°.

Step 2 - Demonstration

  • Teacher draws multiple examples on the board to show that the angle at the center is always twice the angle at the circumference.
    Step 3 - Practice
  • Students are asked to construct similar diagrams and calculate angles at the center and on the circumference based on the theorem.

NOTE ON BOARD:

  • The angle at the center is twice the angle at the circumference subtended by the same chord.

EVALUATION (5 exercises):

  1. If ∠AOC = 50°, what is ∠ABC?
  2. If ∠ABC = 25°, what is ∠AOC?
  3. Draw a circle and label an angle at the center and at the circumference.
  4. How can we prove that the angle at the center is twice the angle at the circumference?
  5. Solve for ∠ABC if ∠AOC = 120°.

CLASSWORK (5 questions):

  1. If ∠AOC = 60°, what is ∠ABC?
  2. Calculate ∠DEF if ∠DGC = 80°.
  3. Draw a circle showing an angle at the center and on the circumference.
  4. If ∠AOC = 90°, what is ∠ABC?
  5. Show that the angle at the center is twice the angle at the circumference using a diagram.

ASSIGNMENT (5 tasks):

  1. Prove that the angle at the center is twice the angle at the circumference.
  2. Draw a diagram showing this theorem in action.
  3. If ∠AOC = 100°, calculate ∠
  4. Give real-life examples of when this theorem might be applied.
  5. Explain why this theorem holds true.

 

PERIOD 3 & 4: Angles in the Same Segment

PRESENTATION: Step 1 - Introduction to Angles in the Same Segment

  • Teacher explains that angles subtended by the same chord in the same segment of the circle are equal.
  • Teacher illustrates this by drawing a circle and labeling two angles subtended by the same chord in different places along the circumference.
    Step 2 - Demonstration
  • Teacher explains and proves that ∠ABC = ∠DEF if both are subtended by the same chord AC.
    Step 3 - Practice
  • Students are asked to draw their own circles and identify angles in the same segment, measuring them for consistency.

NOTE ON BOARD:

  • Angles in the same segment of a circle are equal.

EVALUATION (5 exercises):

  1. If ∠ABC = 40° and ∠DEF = 40°, what can you deduce about the chord?
  2. Draw two angles in the same segment and show that they are equal.
  3. Prove that the angles in the same segment are equal.
  4. Why do you think the angles are equal in the same segment?
  5. Solve for ∠DEF if ∠ABC = 70°.

CLASSWORK (5 questions):

  1. Draw a circle and label two angles in the same segment.
  2. Prove that angles in the same segment are equal by using a diagram.
  3. If ∠A = ∠B, and both subtend the same chord, what does that imply?
  4. If ∠XYZ = 80° and ∠ABC = 80°, are they in the same segment?
  5. Solve for ∠DEF if ∠ABC = 50°.

ASSIGNMENT (5 tasks):

  1. Draw a diagram showing two angles in the same segment and prove that they are equal.
  2. If ∠ABC = 40° and ∠DEF = 40°, explain why they must be in the same segment.
  3. What would happen if the angles were not equal in the same segment?
  4. Can you find any real-world applications where this rule might be useful?
  5. Prove that angles in the same segment are always equal using a different example.

 

PERIOD 5: Opposite Angles of a Cyclic Quadrilateral and Tangent to a Circle

PRESENTATION: Step 1 - Opposite Angles of a Cyclic Quadrilateral

  • Teacher explains that opposite angles of a cyclic quadrilateral are supplementary (they add up to 180°).
  • Teacher draws a cyclic quadrilateral and demonstrates this theorem with examples.
    Step 2 - Tangent to a Circle
  • Teacher demonstrates the concept that the radius at the point of contact with the tangent is perpendicular to the tangent line.
  • Teacher draws a circle with a tangent at a point and shows the right angle formed with the radius.

EVALUATION (5 exercises):

  1. If the sum of two opposite angles of a cyclic quadrilateral is 180°, what are the angles?
  2. Prove that opposite angles of a cyclic quadrilateral are supplementary.
  3. Draw a circle and label the tangent and radius, showing the right angle.
  4. If ∠ABC and ∠DEF are opposite angles in a cyclic quadrilateral, and ∠ABC = 110°, find ∠
  5. Explain why the radius of a circle is perpendicular to the tangent at the point of contact.

 

CLASSWORK (5 questions):

  1. Prove that opposite angles of a cyclic quadrilateral are supplementary.
  2. Draw a diagram showing a tangent to a circle and explain why it is perpendicular to the radius.
  3. If ∠PQR = 90° and ∠RST = 90°, are they opposite angles of a cyclic quadrilateral?
  4. What is the relationship between the radius and the tangent of a circle?
  5. Solve for ∠XYZ in a cyclic quadrilateral if the opposite angle is 140°.

 

ASSIGNMENT (5 tasks):

  1. Prove that opposite angles of a cyclic quadrilateral always add up to 180°.
  2. Draw a diagram showing a cyclic quadrilateral and prove the opposite angles are supplementary.
  3. Explain why the radius of a circle is always perpendicular to the tangent.
  4. If ∠ABC = 50° in a cyclic quadrilateral, what is ∠DEF?
  5. Find the sum of opposite angles in a cyclic quadrilateral and show that it equals 180°.