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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

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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 =
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

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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

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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

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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

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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

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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

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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 = 360 4x
4x = 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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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

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MINI-PLENARY 2 – TICK OR TRASH
Click on the link and download the worksheet. Answers are provided. 19

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LINK BACK TO OBJECTIVES
Accurately calculate missing angles using circle theorems.
What grade are we working at? 20

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PLENARY ACTIVITY 1 T H S E R M A U B 21

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. . O a
110°
Find the value of a
55°

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

25. .
88° e O
Find the value of e
44°

26. . d
18° O
Find the value of d
36°

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

28. . O
40° a
Find the value of a
50°

29. . d O
37°
Find the value of d
53°

30. . O e
Find the value of e
45°

31. . O
65° a
Find the value of a
25°

32. . d O
20°
Find the value of d
70°

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

34. . O a
75°
Find the value of a
15°

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

36. . O a
48°
Find the value of a
42°

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

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

39. . O
56° f
Find the value of f
56°

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

41. . O
61° f
Find the value of f
61°

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

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

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

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

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

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

48. . CIRCLE THEOREMS PLT Skills Which ones are you using?
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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

49. 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?

50. 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

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

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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

53. Which ones are you using?
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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

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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

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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

56. Which ones are you using?
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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

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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

58. Which ones are you using?
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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

59. Which ones are you using?
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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

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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

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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

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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

63. Which ones are you using?
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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

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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

65. 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

66. . 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