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Math lesson plan · 10th grade (Year 11 in England) · 60 minutes

Circle theorems: the alternate segment theorem

Geometry

CCSS HSG.C.A.2England: GCSE Higher Geometry

Objective and success criteria

Students will: Circle theorems: the alternate segment theorem.

  • I can answer questions on today's skill on my own, without help.
  • I can explain each step and say why it works.
  • I can check my answer and spot a mistake.

Before and after

Needs first: Circle theorems: tangents and radii

Leads to: the next skill in this area.

Materials: Rulers and protractors, Squared or dotted paper, Shape cut-outs, Mini whiteboards and pens, Printed student sheet (below).

Warm-up · 6 min

  • Quick retrieval from earlier learning (Circle theorems: tangents and radii). Students answer on mini whiteboards.
  • Show answers together and note who needs support in the main task.
  1. PA and PB are tangents from P to a circle with centre O, touching it at A and B. ∠APB = 31°. Find ∠AOB. → 149Prompt: A tangent meets the radius at 90°.
  2. Two tangents are drawn from point P to a circle, touching it at A and B. PA = (2x + 28) cm and PB = (3x + 16) cm. Find x. → 12Prompt: Tangents from the same point are equal in length.
  3. A tangent from P touches a circle with centre O at T. The radius is 6 cm and OP = 10 cm. Find the length PT. → 8Prompt: Triangle OTP has a right angle at T.

Hook · 6 min

  • Show a photo of a building, a tile pattern or a road sign and ask what shapes and angles students can spot.
  • Share the objective: Circle theorems: the alternate segment theorem. Read the success criteria together.

I do: model it · 12 min

  • Work through the example out loud, saying why you do each step, not just what you do.
  • Example: A tangent touches a circle at A. Chord AB makes an angle of 61° with the tangent. C is a point on the circle in the other segment. Find ∠ACB.
  • Step 1: ∠ACB = 61°
  • Answer: 61
  • Pause after each step and ask a student to say what comes next.

We do: guided practice · 12 min

  • Do these together. Students try each step on whiteboards before you reveal it.
  • Use the hints as prompts rather than giving the answer.
  1. A tangent touches a circle at A. Chord AB makes an angle of 66° with the tangent. C is a point on the circle in the other segment. Find ∠ACB. → 66Prompt: The angle between a tangent and a chord equals the angle in the alternate segment.
  2. A tangent touches a circle at A. Triangle ABC is drawn in the circle. The angle between the tangent and AB is 50°, and ∠ABC = 33°. Find ∠BAC. → 97Prompt: Use the alternate segment theorem to find ∠ACB.

You do: independent practice · 17 min

  • Students work on the student sheet. Circulate and check the first two answers of every student early.
  • Students who finish: write a question of their own for a partner, with the answer.

The 8 questions are on the student sheet below, with answers in the key.

Exit ticket · 7 min

  • Two questions on their own. Sort answers into got it / nearly / not yet to plan the next lesson.
  1. A tangent touches a circle at A. Triangle ABC is drawn in the circle. The angle between the tangent and AB is 54°, and ∠ABC = 60°. Find ∠BAC. → 66Prompt: Use the alternate segment theorem to find ∠ACB.
  2. A tangent touches a circle at A. Chord AB makes an angle of 68° with the tangent. C is a point on the circle in the other segment. Find ∠ACB. → 68Prompt: The angle between a tangent and a chord equals the angle in the alternate segment.

Watch out for

  • Thinking a square turned on its corner is no longer a square.
  • Reading the wrong scale on a protractor.

Support and stretch

Support

  • Review first: Circle theorems: tangents and radii.
  • Use concrete materials: rulers and protractors.
  • Give the first step of each question, or a partly worked example to finish.
  • Pair with a confident partner for the guided practice.

Stretch

  • Explain a common mistake in writing and how to fix it.
  • Write a word problem that uses today's skill, then solve a partner's.

Homework and more practice

Free online practice with hints and printable worksheets for this skill: eurekami.com/math/grade-10.

Free lesson plan from eurekami.com/lesson-plan-builder

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Circle theorems: the alternate segment theorem

Name

Date

Score/ 8

  1. Question 1

    A tangent touches a circle at A. Chord AB makes an angle of 37° with the tangent. C is a point on the circle in the other segment. Find ∠ACB.

    Answer

  2. Question 2

    A tangent touches a circle at A. Triangle ABC is drawn in the circle. The angle between the tangent and AB is 48°, and ∠ABC = 60°. Find ∠BAC.

    Answer

  3. Question 3

    A tangent touches a circle at A. Chord AB makes an angle of 75° with the tangent. C is a point on the circle in the other segment. Find ∠ACB.

    Answer

  4. Question 4

    A tangent touches a circle at A. Triangle ABC is drawn in the circle. The angle between the tangent and AB is 75°, and ∠ABC = 41°. Find ∠BAC.

    Answer

  5. Question 5

    A tangent touches a circle at A. Chord AB makes an angle of 54° with the tangent. C is a point on the circle in the other segment. Find ∠ACB.

    Answer

  6. Question 6

    A tangent touches a circle at A. Triangle ABC is drawn in the circle. The angle between the tangent and AB is 71°, and ∠ABC = 61°. Find ∠BAC.

    Answer

  7. Question 7

    A tangent touches a circle at A. Chord AB makes an angle of 29° with the tangent. C is a point on the circle in the other segment. Find ∠ACB.

    Answer

  8. Question 8

    A tangent touches a circle at A. Triangle ABC is drawn in the circle. The angle between the tangent and AB is 46°, and ∠ABC = 68°. Find ∠BAC.

    Answer

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Answer key: Circle theorems: the alternate segment theorem

  1. 1. 37

  2. 2. 72

  3. 3. 75

  4. 4. 64

  5. 5. 54

  6. 6. 48

  7. 7. 29

  8. 8. 66