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Circle Theorems: Angles and Segments

Enhance mathematical skills and knowledge by accurately calculating missing angles and identifying properties of circles using circle theorems. Grades B/A. Explore examples and tasks to practice applying circle theorems.

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Circle Theorems: Angles and Segments

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  1. Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager Real life cross/curricular links? Where are we in our journey? Which ones are you using? PLT Skills LESSON OBJECTIVES Always aim high! We are learning to: • Enhance Mathematical basic skills knowledge. (Which PLT skills?) • Accurately calculate missing angles using circle theorems. (Grades B/A). AUTHOR www.mistrymaths.co.uk

  2. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager STARTER PLT Skills TASK Distance = 615miles Speed = 82mph Time = ……….minutes 2) 3) 1) Write in standard form: 2.61 3cm 450 0 = 2.61 x 10 6cm 615 = = 7.5hours 82 N 11cm Area shaded black? Write the bearing. Solve: 4x + 3y = 10 5x - 2y = 24 5) 4) Area of circle – Area of triangle 095° Π x r x r - = - 3.14 x 5.5 x 5.5 = = - 95.0 9 8x + 6y = 20 6) 15x - 6y = 72 + 5m 2 86cm 92 = 23x 5m Π x d 360 3.14 x 10 360 b x h 6 x 3 x 4 = Work out the perimeter 7 9 2 2 4 4 16 + 3y = 10 + r + r + 5 + 5 3y = -6 = EXTENSION -2 y = 17.85m = = 51.4° Heptagon → Find the sum of one exterior angle in an regular nonagon and one exterior angle in a regular heptagon? = 40° Nonagon →

  3. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager o o xo yo yo xo PLT Skills CIRCLE THEOREMS INTRODUCTION xo xo o o a + b = 180° DIAMETER ao x + y = 180° 2xo 2xo xo bo yo Angle made by a diameter is 90°. Angle at centre is twice the angle at the circumference. Angle at centre is twice the angle at the circumference. Opposite angles in a cyclic quadrilateral add up to 180° yo xo RADIUS yo RADIUS xo Alternate segment theorem. Angles subtended by an arc or a chord in the same segment are equal. Angle between a tangent and a radius make 90°

  4. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager n° g° 60o 30° j° q° 45o p° h° o k° 70o i° PLT Skills CIRCLE THEOREMS EXAMPLES 4) 2) 3) 1) fo 30° 42° o o 57° 82° ao 86° bo eo 98° 123° 31° 60° 43° 6) 5) 7) 90° co 31° l° 38° 38° 20° 45° 90° 42° m° 38° do 90° 75° 60° 60° Angles in a triangle add to 180° Angles on a straight line add to 180°

  5. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills CIRCLE THEOREMS TASK 1 – ANGLE AT THE CENTRE (GRADE B) Find the angles marked in each of these circles with centre O. 3) 4) 2) 1) a 44° 70° . . . . . . . 55° d . c O O 42° O 22° b 110° 84° O 140° 5) 8) 6) 7) e 113° 192° 43° f 11° O 11° 158° O O 226° 96° O 86° 79° 47° g h

  6. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills CIRCLE THEOREMS TASK 2 – ANGLE MADE BY DIAMETER (GRADE B) Find the angles marked in each of these circles with centre O. 3) 4) 2) 1) d 26° . . . . . . . . 72° 60° 30° a O O O O b 42° 64° 48° c 18° 5) 8) 6) 7) 30° 30° h g O 60° 45° 40° f O O O 45° e 40° 80° 50°

  7. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills CIRCLE THEOREMS TASK 3 – CYCLIC QUADRILATERALS (GRADE B) Find the angles marked in each of these circles with centre O. 3) 4) 2) 1) e f 109° a 103° 102° 100° 100° 88° . . . . . . . . i b 103° O O O 70° 77° 80° O 71° g 110° j 80° 77° 78° d c h 92° 5) 8) 6) 7) 121° 88° q 59° 87° r 92° k t 119° 92° 108° 119° O 104° O 76° O O n 61° 88° 88° s 93° p o 121° 72° 108° l m 61° 119°

  8. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills CIRCLE THEOREMS TASK 4 – ANGLE MADE BY TANGENT AND RADIUS (GRADE B) Find the angles marked in each of these circles with centre O. 3) 2) 1) a 35° b . . . . . . 49° O O 55° O 41° d c 120° 65° 25° 60° 4) 5) 6) x 18° O 72° 3x + x + 90 + 90 = 360 4x 4x + 180 = 360 O 4x = 180 O e O x = 45° 68° 22° 3x 4x + x + 90 = 180 f 135° 136° 5x + 90 = 180 5x = 90 45° x x = 18°

  9. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills CIRCLE THEOREMS TASK 5 – ANGLES IN THE SAME SEGMENT (GRADE B) Find the angles marked in each of these circles with centre O. 3) 2) 1) 31° b 31° . . . . . . d 51° 13° O O 9° a 13° e 9° O 51° 87° 56° 56° 42° c 4) 5) 6) 44° 72° 18° g h 18° 15° O O O 44° 103° 62° 44° f 62°

  10. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills CIRCLE THEOREMS TASK 6 – ALTERNATE SEGMENT THEOREM (GRADE A) Find the angles marked in each of these circles with centre O. 3) 2) 1) g 74° 59° 47° 36° 82° . . . . . . a 73° h O O 59° O 85° 25° 74° c 59° 73° i e 85° 36° d f 82° b 4) 5) 6) k m q 82° 72° 72° l r s = t t 36° 49° 49° O O O 82° s 72° 49° p 72° j 57° n 57° t° 114° 66°

  11. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills CIRCLE THEOREMS TASK 7 – MIXTURE OF THEOREMS (GRADE B) Find the angles marked in each of these circles with centre O. 3) 2) 1) 106° 113° a . . . . . . 60° 50° O O O O b O 74° O 40° 120° d 67° c 4) 5) 6) e 32° 6° 37° g 37° 6° 88° 53° 37° 60° i 88° f h j 32°

  12. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills CIRCLE THEOREMS TASK 7 – MIXTURE OF THEOREMS (GRADE B) Find the angles marked in each of these circles with centre O. 9) 8) 7) 121° . . . . . . 45° 104° 62° a 59° 121° 31° O b O O 76° d c 10) 11) 12) 2x 20° q 112° 70° r e 34° 34° 7x 33° O O O 112° 7x + 2x + 90 = 180 p 34° 98° 98° 9x + 90 = 180 n 49° 33° 9x = 90 x = 10°

  13. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills CIRCLE THEOREMS EXTENSION 1 – MIXTURE OF THEOREMS (GRADE A) Find the angles marked in each of these circles with centre O. 3) 2) 1) 33° 20° 20° 129° 115° 55° 115° a c 115° 55° b 51° d 160° 48° . . . . . . 17° 130° 111° 130° e 18° f O 25° O O 230° 230° 18° 4) 5) 6) 34° k 38° 26° 106° h 116° O g 64° 128° 73° O O 36° 73° 26° 146° i 17° j 74°

  14. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills CIRCLE THEOREMS EXTENSION 1 – MIXTURE OF THEOREMS (GRADE A) Find the angles marked in each of these circles with centre O. 9) 8) 7) e c a 56° 68° 124° 80° . . . . . . 56° O O 10° d 160° 112° 56° f 124° 100° 80° b 56° 10) 11) 12) 55° h g 35° O 6° O O 35° 72° 84° 84° 108° 72° 36° i 36°

  15. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills CIRCLE THEOREMS EXTENSION 2 – EXAM QUESTIONS (GRADES B-A) Find the angles marked in each of these circles with centre O. 2) 1) Prove that triangle QSR is isosceles (3 marks) Work out the value of each letter (3 marks) . . . . Therefore, QSR is isosceles O q O 70° 140° p Q 40° 80° 50° 50° 40° 40° R S Work out the value of each letter (4 marks) 3) 4) Work out the value of each letter (2 marks) 62° 28° y 25° 124° x O 140° 28° O 40° 96° 65° 65° 42° x y

  16. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills CIRCLE THEOREMS EXTENSION 2 – EXAM QUESTIONS (GRADES B-A) Find the angles marked in each of these circles with centre O. 6) 5) Work out the value of the letter (2 marks) Work out the value of each letter (4 marks) 32° 116° . . . . b 32° a O O 32° 32° c 74° 42° 116° 32° 58° 74° Work out the value of each letter (2 marks) 7) 8) Work out the value of the letter (2 marks) 40° x 40° 60° X = 120° x 30° O O 30° 60° 140° y

  17. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills CIRCLE THEOREMS EXTENSION 2 – EXAM QUESTIONS (GRADES B-A) Find the angles marked in each of these circles with centre O. 9) 10) 32° 116° 3x Alternate angles are equal 45° . . 32° 32° O O 64° 15° 30° 2x a 135° 9x 32° b 120° 15° Work out the value of the letter (5 marks) Show that x = 15° and work out the missing letter (6 marks) 9x + 3x = 180 12x = 180 x = 15°

  18. MINI-PLENARY 1 – CIRCLE THEOREMS CATCHPHRASE

  19. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills CIRCLE THEOREMS MINI-PLENARY 2 – TICK OR TRASH Click on the link and download the worksheet. Answers are provided. http://www.tes.co.uk/teaching-resource/Circle-Theorems-Tick-or-Trash-angles-6328139/

  20. Which ones are you using? Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager • LINK BACK TO OBJECTIVES • Accurately calculate missing angles using circle theorems. What grade are we working at? DISCOVERY PLT Skills

  21. Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager A H S M T B S E S U R T PLT Skills Which ones are you using? CIRCLE THEOREMS PLENARY ACTIVITY 1

  22. A C E B D F H J G I K M O L N P R T Q S U W Y V X

  23. a 110° Find the value of a . O 55°

  24. 66° a Find the value ofa . O 132°

  25. 88° e O Find the value ofe . 44°

  26. d 18° O Find the value of d . 36°

  27. 52° a Find the value ofa . O 104°

  28. 40° a Find the value of a . O 50°

  29. d O 37° Find the value ofd . 53°

  30. O e Find the value ofe . 45°

  31. 65° a Find the value ofa . O 25°

  32. d O 20° Find the value ofd . 70°

  33. a b 110° 95° Find the values of a and b . O a = 85° and b = 70°

  34. a 75° Find the value of a . O 15°

  35. a b 120° 105° Find the values of a and b . O a = 75° and b = 60°

  36. a 48° Find the value of a . O 42°

  37. a b 130° 115° Find the values of a and b . O a = 65° and b = 50°

  38. 33° b a 15° Find the values of a and b . O a = 15° and b = 33°

  39. O 56° f Find the value of f . 56°

  40. 42° b a 19° Find the values of a and b . O a = 19° and b = 42°

  41. O 61° f Find the value of f . 61°

  42. 56° b a 22° Find the values of a and b . O a = 22° and b = 56°

  43. 26° 75° d e Find the values of d and e . O d = 75° and e = 26°

  44. 85° a 71° b Find the values of a and b . O a = 71° and b = 85°

  45. 19° 69° d e Find the values of d and e . O d = 69° and e = 19°

  46. 64° a 75° b Find the values of a and b . O a = 75° and b = 64°

  47. 15° 86° d e Find the values of d and e . O d = 86° and e = 15°

  48. Team Worker Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager PLT Skills Which ones are you using? CIRCLE THEOREMS PLENARY ACTIVITY 2 – GRADE A* EXAM STYLE QUESTION PT is a tangent to a circle with centre O. AB are points on the circumference. Angle PBA is x. . B 2x 90 - x x x A O (a) Write down the value of angle AOP (b) Calculate the angle OPA in terms of x. The triangle OPA is isosceles since sides OP and OA are both radii. T 2x P Angles in a triangle add up to 180° and the base angles are equal. 180° - 2x OPA = (c) Prove that the angle APT is equal to the angle PBA 2 90° - x OPA = Angle APT = Angle PBA because of the alternate segment theorem

  49. What grade are we working at? Where are we in our journey? What have you learnt? Draw your brain In your brain, write or draw everything you can remember about calculating missing angles using circle theorems. It can be a skill or a reflection, or something else that might be prominent in your brain.

  50. Team Worker Positive Thinker Creative Entrepreneur Independent Learner Reflective Learner Responsible Citizen SELF ASSESSMENT Enterprise Skills Which ones are you using? Plenary Activity How well do you understand the task? . I fully understand I don’t understand I nearly understand www.mistrymaths.co.uk

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