Mathematics - Senior Secondary 2 - Circle theorems

Circle theorems

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TERM: 2ND TERM

WEEK: 4

Class: Senior Secondary School 2
Age: 16 years
Duration: 40 minutes of 5 periods
Subject: Mathematics
Topic: Circle Theorems
Focus: Proofs of Various Circle Theorems
Specific Objectives:
By the end of the lesson, students should be able to:

  1. Prove the theorem that the angle subtended by an arc at the center of a circle is twice the angle subtended at the circumference.
  2. Prove the theorem that angles in the same segment of a circle are equal.
  3. Prove that the angle in a semi-circle is a right angle.
  4. Prove the theorem that opposite angles of a cyclic quadrilateral are supplementary.
  5. Prove that a tangent to a circle is perpendicular to the radius at the point of contact.

Instructional Techniques:
• Question and answer
• Guided demonstration
• Discussion
• Practice exercises
• Analogy and real-life connections

Instructional Materials:
• Models of circle theorems (e.g., circular diagrams, compasses, protractors)
• Whiteboard and markers
• Worksheets for practice
• Visual aids to explain circle theorems

PERIOD 1 & 2: Proof of Theorem 1: The Angle Which an Arc Subtends at the Center is Twice the Angle Subtended at the Circumference

Presentation:

Step

Teacher’s Activity

Student’s Activity

Step 1 - Introduction

Introduces the concept of angles in a circle. Reviews terms like "center," "circumference," and "arc."

Students listen and ask questions.

Step 2 - Theorem Explanation

Explains the theorem: The angle subtended by an arc at the center of a circle is twice the angle subtended at the circumference. Uses a diagram for illustration.

Students observe and take notes.

Step 3 - Proof

Demonstrates the proof by constructing a circle and using a diagram. Mentions using basic geometric principles to prove the relationship.

Students follow along with the proof process and take notes.

Step 4 - Guided Practice

Provides students with an arc and asks them to identify and calculate the central and inscribed angles, proving the theorem.

Students work on exercises under teacher guidance.

Note on Board:
Theorem 1: The angle subtended at the center of a circle by an arc is twice the angle subtended at the circumference.
Proof: Use geometric properties (e.g., isosceles triangles, angles at the center and circumference).

Evaluation (5 exercises):

  1. Prove that the angle subtended by an arc at the center is twice the angle at the circumference.
  2. Draw a circle and mark the center and circumference. Identify angles that satisfy the theorem.
  3. How does this theorem apply to a chord of a circle?
  4. Find the angle subtended at the center and at the circumference for a given arc in a circle.
  5. What is the relationship between the angle at the center and at the circumference for a full circle?

Classwork (5 questions):

  1. In a circle with center O, if the angle at the center is 60°, what is the angle at the circumference?
  2. Given a circle, if the angle at the circumference is 40°, what is the angle at the center?
  3. What would be the angle at the center if the angle at the circumference is 50°?
  4. Prove that if an arc subtends an angle of 30° at the circumference, the angle at the center is 60°.
  5. How does the length of the arc affect the angles in the circle?

Assignment (5 tasks):

  1. Draw a circle and label the center, arc, and circumference. Mark all the relevant angles.
  2. Prove the relationship between the angle at the center and the angle at the circumference using specific numbers.
  3. Find an example where the theorem is applied in real life.
  4. Calculate the angle at the center given the angle at the circumference.
  5. Prove that the central angle is always twice the inscribed angle for any circle.

PERIOD 3 & 4: Proof of Theorem 2: Angles in the Same Segment Are Equal

Presentation:

Step

Teacher’s Activity

Student’s Activity

Step 1 - Introduction

Introduces the concept of segments in a circle and reviews what makes up a segment.

Students listen and engage in the discussion.

Step 2 - Theorem Explanation

Explains the theorem: Angles in the same segment of a circle are equal. Provides a diagram to show angles subtended by the same chord.

Students observe and take notes.

Step 3 - Proof

Demonstrates the proof using a circle, labeling the chord and angles in the segment. Uses isosceles triangles and basic geometry to prove the angles are equal.

Students follow along with the proof process.

Step 4 - Guided Practice

Provides students with different segments of a circle and asks them to prove that the angles are equal using the theorem.

Students work on exercises in pairs or individually.

Note on Board:
Theorem 2: Angles in the same segment of a circle are equal.
Proof: Use symmetry, congruence of triangles, and the properties of angles in a circle.

Evaluation (5 exercises):

  1. Prove that angles in the same segment of a circle are equal.
  2. Given a circle with a chord, prove that two angles subtended by the chord are equal.
  3. What is the significance of this theorem in geometry?
  4. How can you use this theorem to find unknown angles in a circle?
  5. Can you think of a real-world application of this theorem?

Classwork (5 questions):

  1. In a circle, prove that two angles subtended by the same chord are equal.
  2. Given a circle, find the missing angle if two angles in the same segment are 45° and 45°.
  3. In a circle, if one angle is 60° and another is 60°, are they in the same segment? Prove your answer.
  4. How can you prove that two angles formed by the same chord are equal?
  5. How can you use the segment theorem to solve angle problems in a circle?

Assignment (5 tasks):

  1. Prove that two angles subtended by the same chord in a circle are equal.
  2. Solve for the missing angle in a circle using the segment theorem.
  3. What happens if the angles in the same segment are not equal?
  4. Create a real-life situation where the angle segment theorem applies.
  5. Find the measure of angles in a given circle segment.

PERIOD 5: Proof of Theorem 3: The Angle in a Semi-circle is a Right Angle

Presentation:

Step

Teacher’s Activity

Student’s Activity

Step 1 - Introduction

Introduces the concept of a semicircle and reviews the properties of diameters and chords.

Students listen and ask questions.

Step 2 - Theorem Explanation

Explains that the angle subtended by a diameter in a semicircle is always a right angle. Uses a diagram to show the situation.

Students observe the diagram and take notes.

Step 3 - Proof

Demonstrates the proof using geometric properties, such as the isosceles triangle and the right-angle property of a semicircle.

Students follow along with the proof process.

Step 4 - Guided Practice

Provides several semicircle examples for students to prove that the angle is always 90° in each case.

Students work on exercises individually.

Note on Board:
Theorem 3: The angle in a semicircle is always a right angle.
Proof: Use basic geometric principles (right angle in a semicircle).

Evaluation (5 exercises):

  1. Prove that the angle in a semicircle is a right angle.
  2. Given a semicircle, find the angle subtended at the circumference.
  3. What makes the angle in a semicircle a right angle?
  4. Use the semicircle theorem to solve an angle problem.
  5. How does this theorem apply to other geometrical shapes?

Classwork (5 questions):

  1. In a semicircle, prove that the angle at the circumference is 90°.
  2. If the angle at the center of a circle is 60°, what is the angle at the circumference for a semicircle?
  3. Draw a semicircle and label the diameter, center, and angle at the circumference.
  4. Find the angle in a semicircle if the diameter is given.
  5. How does the semicircle theorem help solve problems in circular geometry?

Assignment (5 tasks):

  1. Prove that the angle in a semicircle is always 90°.
  2. Find the angle in a semicircle given a certain measurement.
  3. Draw a semicircle and calculate the angle at the circumference.
  4. Apply the semicircle theorem to a real-life problem.
  5. How can you use this theorem in geometric proofs?