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 helpful-report@pixl.org.uk

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

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

Key Circle Vocabulary Radius Diameter Circumference Arc Sector Tangent Chord Segment

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

Circle Theorems Student Sheet 2

'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°

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

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

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°

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

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

Student Sheet 3

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)

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

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

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

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

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

Problem Solving - Solutions Angle a = 65 Angles in a triangle add up to 180 b = 180-45-65 = 70 Opposite angles in a cyclic quadrilateral add up to 180 d = 180-45 = 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

Problem Solving - Solutions

Exam Questions – Specimen Papers - 1 Student Sheet 4

Exam Questions – Specimen Papers

Exam Questions – Specimen Papers

Exam Questions – Specimen Papers

Exam Questions – Specimen Papers - 2 Student Sheet 5

Exam Questions – Specimen Papers

Exam Questions – Specimen Papers

Exam Questions – Specimen Papers