1. Grade 8
Circle Theorems
Apply and prove the standard circle theorems concerning angles, radii, tangents and chords, and use them to prove related results
If you have any questions regarding these resources or come across any errors, please contact

2. Lesson Plan Lesson Overview Progression of Learning
Objective(s)
Apply and prove the standard circle theorems concerning angles, radii, tangents and chords, and use them to prove related results
Grade 8
Prior Knowledge
Circle terminology
Basic angle reasoning
Duration
Students need to memorise the theorems so if this is the first lesson the content will need to be revisited. The complexity is when several theorems and angle rules are combined and students need complete a multi staged answer in order to obtain the required angle. Often language can be confusing here. For each theorem the slide shows the full theorem (in a sentence with all mathematical words) and a simplified version (in yellow) which is the bare minimum required to get full marks in the exam. 60 minutes are required but subsequent practice if this is first encounter.
Resources
Print slides: 3, 6, 13, 22, 26
Equipment
Progression of Learning
What are the students learning?
How are the students learning? (Activities & Differentiation)
Circle terminology specific to this topic
Give students slide 3 printed. To work independently to match the word with the part of the circle (content should be recap of prior knowledge). Use slide 4 to review the correct matching. 10
The language of the theorems and associated visuals
Give students slide 6 printed. On this they need to write each theorem next to the appropriate image. Use slides 7 to 12 to explore each theorem. For each theorem the slide shows the full theorem (in a sentence with all mathematical words) and a simplified version (in yellow) which is the bare minimum required to get full marks in the exam. Share this with students and get them to record the version most appropriate 15
Finding missing angles using circle theorems
Give students slide 13. For each missing angle must state the numerical value and the associated theorem as the reason. Use slides 14 to 21 to explain each answer. Spend more time collectively reviewing the questions which involve multi stages. 20
Finding missing angles using circle theorems in exam questions (from specimen papers)
Give students slide 22 and 28. This includes 6 exam questions related to objective. Students need to use notes from lesson to answer the questions. Ensure that all steps are shown. Relate to mark scheme to show how the marks are allocated.
Next Steps
Combined problems with trigonometry
Assessment
PLC/Reformed Specification/Target 8/Geometry & Measure/Circle Theorems

3. Key Circle Vocabulary Radius Diameter Circumference Arc Sector Tangent
Chord
Segment
Student Sheet 1

4. Key Circle Vocabulary Radius Diameter Circumference Arc Sector Tangent
Chord
Segment

5. The following six Circle Theorems MUST be learned in order to explain each part of the calculations.

6. Circle Theorems
Student Sheet 2

7. 'Subtended' is just a technical way of saying 'made'
Theorem 1: The angle subtended at the circumference in a semi circle
is a right angle
'Subtended' is just a technical way of saying 'made'
Angle in semi circle 90°

8. Angle at centre twice the angle at circumference
Theorem 2: The angle subtended by an arc at the centre is twice the angle subtended at the circumference
An arc is just a connected section of the circumference
Angle at centre twice the angle at circumference

9. Angles at circumference equal
Theorem 3: Angles at the circumference are equal
Angles at the circumference are equal, if the angles stand on the same arc
Angles at circumference equal

10. The opposite angles of a cyclic quadrilateral add up to 𝟏𝟖𝟎°
Theorem 4: Cyclic quadrilateral
The opposite angles of a cyclic quadrilateral add up to 𝟏𝟖𝟎°
The exterior angle of a cyclic quadrilateral equals the opposite interior angle
Cyclic quadrilateral opposite angles add up to 180°

11. Tangent meets radius at 90° Tangents from point are equal
Theorem 5: Tangent to a circle
Tangent meets radius at 90°
Tangents from point are equal

12. Alternate segment theorem
Theorem 6: Alternate segment
The angle between a tangent and a chord is equal to the angle at the circumference in the alternate segment.
Alternate segment theorem

13. Student Sheet 3

14. Find the missing angles below given reasons in each case.
You do…
85o
110o x y
70o
135o p r q
Find the missing angles below given reasons in each case.
angle x =
angle y =
angle p =
angle q =
angle r =
180 – 85 = 95o (cyclic quad)
180 – 135 = 45o (straight line)
180 – 110 = 70o (cyclic quad)
180 – 70 = 110o (cyclic quad)
180 – 45 = 135o (cyclic quad)

15. You do… Q O T P yo xo angle w = angle x = angle y = angle z = 98o zo
PQ and PT are tangents to a circle with centre O. Find the unknown angles giving reasons. yo Q xo O
angle w =
angle x =
angle y =
angle z =
98o zo wo T P

16. You do… Q O T P yo xo angle w = angle x = angle y = angle z =
PQ and PT are tangents to a circle with centre O. Find the unknown angles giving reasons. yo Q xo O
angle w =
angle x =
angle y =
angle z =
90o (tan/rad)
98o
90o (tan/rad)
49o (angle at centre)
360o – 278 = 82o (quadrilateral) zo wo T P

17. You do… Q O T P yo angle w = angle x = angle y = xo 80o wo 50o
PQ and PT are tangents to a circle with centre O. Find the unknown angles giving reasons. Q O yo
angle w =
angle x =
angle y = xo
80o wo
50o T P

18. You do… Q O T P yo angle w = angle x = angle y = 90o (tan/rad) xo
PQ and PT are tangents to a circle with centre O. Find the unknown angles giving reasons. Q O yo
angle w =
angle x =
angle y =
90o (tan/rad) xo
180 – 140 = 40o (angles sum tri)
50o (isos triangle)
80o wo
50o T P

19. Problem Solving
Calculate the angles a, b, c, and d

20. Problem Solving - Solutions
Angle a = 65
Angles in a triangle add up to 180
b = = 70
Opposite angles in a cyclic quadrilateral add up to 180
d = = 135
Tangents which meet at the same point are the same length. Angles in a triangle add up to 180.
c = 180 – 65 – 65 = 50

21. Problem Solving - Solutions

22. Exam Questions – Specimen Papers - 1
Student Sheet 4

23. Exam Questions – Specimen Papers

24. Exam Questions – Specimen Papers

25. Exam Questions – Specimen Papers

26. Exam Questions – Specimen Papers - 2
Student Sheet 5

27. Exam Questions – Specimen Papers

28. Exam Questions – Specimen Papers

29. Exam Questions – Specimen Papers