Which ones are you using?

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Which ones are you using? Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager Team Worker CIRCLE THEOREMS PLT Skills Which ones are you using? EXTENSION 2 – EXAM QUESTIONS (GRADES B-A) Find the angles marked in each of these circles with centre O. 9) 10) . O . O 32° 116° 3x Alternate angles are equal 45° 32° 32° 64° 15° 30° a 135° 2x 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° 17

MINI-PLENARY 1 – CIRCLE THEOREMS CATCHPHRASE Put your money where your mouth is. 18

Which ones are you using? Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager Team Worker CIRCLE THEOREMS PLT Skills Which ones are you using? 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/ 19

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

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

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

. O a 110° Find the value of a 55°

. O 66° a Find the value of a 132°

. 88° e O Find the value of e 44°

. d 18° O Find the value of d 36°

. O 52° a Find the value of a 104°

. O 40° a Find the value of a 50°

. d O 37° Find the value of d 53°

. O e Find the value of e 45°

. O 65° a Find the value of a 25°

. d O 20° Find the value of d 70°

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

. O a 75° Find the value of a 15°

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

. O a 48° Find the value of a 42°

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

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

. O 56° f Find the value of f 56°

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

. O 61° f Find the value of f 61°

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

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

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

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

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

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

. CIRCLE THEOREMS PLT Skills Which ones are you using? Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager Team Worker CIRCLE THEOREMS PLT Skills Which ones are you using? 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. . P A B T x O 2x 90 - x x (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. 2x 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 Angle APT = Angle PBA because of the alternate segment theorem OPA = 90° - x 48

Where are we in our journey? What grade are we working at? 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. Where are we in our journey? What grade are we working at?

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

Creative Entrepreneur Positive Thinker Creative Entrepreneur SELF ASSESSMENT Independent Learner Reflective Learner Responsible Citizen Team Worker Enterprise Skills Which ones are you using? Plenary Activity WWW (What Went Well) EBI (Even Better If) On your post it notes… Think about how you can improve your work. www.mistrymaths.co.uk

Which ones are you using? Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager Team Worker CIRCLE THEOREMS PLT Skills Which ones are you using? INTRODUCTION 2xo xo o xo 2xo o o DIAMETER bo xo yo ao a + b = 180° x + y = 180° 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° xo yo o RADIUS xo yo Angles subtended by an arc or a chord in the same segment are equal. Alternate segment theorem. Angle between a tangent and a radius make 90° 52

Which ones are you using? Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager Team Worker CIRCLE THEOREMS PLT Skills Which ones are you using? TASK 1 – ANGLE AT THE CENTRE (GRADE B) Find the angles marked in each of these circles with centre O. 1) 2) 3) 4) . O . O . O . a 55° c d b 22° 84° O 140° 5) 6) 7) 8) . . e . . 43° f O 11° O O 226° 96° O g h 53

Which ones are you using? Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager Team Worker CIRCLE THEOREMS PLT Skills Which ones are you using? TASK 2 – ANGLE MADE BY DIAMETER (GRADE B) Find the angles marked in each of these circles with centre O. 1) 2) 3) 4) . O . O . O . 26° d 30° a O 42° b c 18° 5) 6) 7) 8) . . . . g h O O f O O e 80° 54

Which ones are you using? Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager Team Worker CIRCLE THEOREMS PLT Skills Which ones are you using? TASK 3 – CYCLIC QUADRILATERALS (GRADE B) Find the angles marked in each of these circles with centre O. 1) 2) 3) 4) . O . O . O . e a 109° f 103° 102° i b O 110° g 80° j c d h 92° 5) 6) 121° 7) 8) . . . O 88° o p q . 87° r k t O 76° O n O s l 108° m 119° 55

Which ones are you using? Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager Team Worker CIRCLE THEOREMS PLT Skills Which ones are you using? TASK 4 – ANGLE MADE BY TANGENT AND RADIUS (GRADE B) Find the angles marked in each of these circles with centre O. 1) 2) 3) . O . O . O a b 55° 41° c d 25° 60° 4) 5) 6) x . . . 4x O O e O 68° 3x f x 56

Which ones are you using? Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager Team Worker CIRCLE THEOREMS PLT Skills Which ones are you using? TASK 5 – ANGLES IN THE SAME SEGMENT (GRADE B) Find the angles marked in each of these circles with centre O. 1) 2) 3) . O . O . O 31° b d a 13° 9° e 56° 87° 42° c 4) 5) 6) . . . 72° g h O O O 103° 44° f 62° 57

Which ones are you using? Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager Team Worker CIRCLE THEOREMS PLT Skills Which ones are you using? TASK 6 – ALTERNATE SEGMENT THEOREM (GRADE A) Find the angles marked in each of these circles with centre O. 1) 2) 3) . O . O g . O 59° 82° 36° 47° a h 85° c 73° i e b f d 4) 5) 6) . m k . q . l r t 49° O O O s 72° j p n 114° 58

Which ones are you using? Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager Team Worker CIRCLE THEOREMS PLT Skills Which ones are you using? TASK 7 – MIXTURE OF THEOREMS (GRADE B) Find the angles marked in each of these circles with centre O. 1) 2) 3) . O . O . O c d 113° 106° 50° b a 120° 4) 5) 6) . O . O . O f g 6° 37° 37° e 32° 88° i j h 59

Which ones are you using? Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager Team Worker CIRCLE THEOREMS PLT Skills Which ones are you using? TASK 7 – MIXTURE OF THEOREMS (GRADE B) Find the angles marked in each of these circles with centre O. 7) 8) 9) O a 31° 121° 76° c d . . . b O O 10) 11) 12) 7x 2x . . . q 49° 98° e r 34° O O O p n 60

Which ones are you using? Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager Team Worker CIRCLE THEOREMS PLT Skills Which ones are you using? EXTENSION 1 – MIXTURE OF THEOREMS (GRADE A) Find the angles marked in each of these circles with centre O. 1) 2) 3) . O 33° . O . O a c b 160° d 48° 17° 130° f e 18° 4) 5) 6) . . . k 26° h g O 73° O 36° O j i 74° 61

Which ones are you using? Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager Team Worker CIRCLE THEOREMS PLT Skills Which ones are you using? EXTENSION 1 – MIXTURE OF THEOREMS (GRADE A) Find the angles marked in each of these circles with centre O. 7) 8) 9) . O . O . O c e a 124° 56° 112° d f 80° b 10) 11) 12) . . . h g O O O 35° 84° 36° i 72° 62

Which ones are you using? Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager Team Worker CIRCLE THEOREMS PLT Skills Which ones are you using? EXTENSION 2 – EXAM QUESTIONS (GRADES B-A) Find the angles marked in each of these circles with centre O. 1) 2) Prove that triangle QSR is isosceles (3 marks) . O . O Work out the value of each letter (3 marks) q p Q 40° 80° 40° S R 3) Work out the value of each letter (4 marks) 4) Work out the value of each letter (2 marks) . 62° . y x O O 40° 65° y x 63

Which ones are you using? Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager Team Worker CIRCLE THEOREMS PLT Skills Which ones are you using? EXTENSION 2 – EXAM QUESTIONS (GRADES B-A) Find the angles marked in each of these circles with centre O. 5) 6) Work out the value of the letter (2 marks) . O . O Work out the value of each letter (4 marks) b a c 116° 58° 74° 7) Work out the value of each letter (2 marks) 8) Work out the value of the letter (2 marks) . . 40° x x O O 30° y 64

Which ones are you using? Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager Team Worker CIRCLE THEOREMS PLT Skills Which ones are you using? EXTENSION 2 – EXAM QUESTIONS (GRADES B-A) Find the angles marked in each of these circles with centre O. 9) 10) . O . O 3x a 2x 9x 32° b Work out the value of the letter (5 marks) Show that x = 15° and work out the missing letter (6 marks) 65

. CIRCLE THEOREMS PLT Skills Which ones are you using? Creative Thinker Effective Participator Independent Enquirer Reflective Learner Self Manager Team Worker CIRCLE THEOREMS PLT Skills Which ones are you using? 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. . P A B T x O (a) Write down the value of angle AOP (b) Calculate the angle OPA in terms of x. (c) Prove that the angle APT is equal to the angle PBA 66