angles and polygons - navigate mathnavigatemath.weebly.com/uploads/7/...anglespolygons... · 2...

www.mathletics.com

Curriculum Ready

Angles

Angles andPolygons

Solutions

13HSERIES TOPIC

1Angles and Polygons SolutionsMathletics Passport © 3P Learning

How does it work? Solutions Angles and Polygons

questions

a

c

a b

b

d

1

2

+XYZ =

+ABC =

+JKL =

+PQR =

+XZY =

+ACB =

+JLK =

+PRQ =

+YXZ =

+BAC =

+KJL =

+QPR =

Sum =

Sum =

Sum =

Sum =

Angle sum of TABC

= +ABC + +ACB + +BAC.

= 180c

Angle sum of TEFG

= +EFG + +EGF + +FEG.

= 180c

` + +BCA + = 180c

` + +BCA = 180c

` +BCA =

` +FEG + + = 180c

` +FEG + = 180c

` +FEG =

Angle sum of a triangle

X

K

J

Q

A B

C

A

85c 42c

105c70c

E F

GBC

P

Q

L

Z Y

50c 40c

50c 60c

80c 80c

180c 180c

90c 32c

35c 38c

55c

70c 105c85c 42c

145c 147c

35c 33c

110c

180c 180c

2 Angles and Polygons SolutionsMathletics Passport © 3P Learning

13HSERIES TOPIC



3

4

5

a

a

a

d

b

b

b

e

c

c

c

f

x =

a =

d =

x =

b =

e =

x =

c =

f =

F

WY

X Z

V

UG

E

75c

88c

x45c

x

66c 43c

35c x

26c

45c 22c

49c60c

a

95c25.5c

56.5c

b

35c c

76.3c

47.6c43.8ce

30.4c53.1c

d

72.5c 127.9c

f

questions

EGF 60c+ = UVW 98c+ = YXZ 41c+ =

47c

59.5c

54.4c

71c

33.5c

21.7c

119c

68.7c

88.6c

13HSERIES TOPIC



+PQR =

+QRS =

+RSP =

+QPS =

Sum =

Angle sum of a quadrilateral

+WXY =

+XYZ =

+YZW =

+XWZ =

Sum =

x = y = z =

v = w = u =

questions

Page 7 questions

50c

60c

110c 98c

80c 55c

90c 125c

80c 82c

360c 360c

4c

18c

75c

15c

6 a

d

b

e

c

f

2u

44c

36c89c

71c5v

w

30c

w

x

x x 2y

3y2z

8z

2z


u

u

u

2 36 44 180

2 80 180

2 100

c c c

c c

c

+ + =

+ =

=

180

3 180

x x x

x

c

c

+ + +=

=

v

v

v

5 89 71 180

5 160 180

5 20

c c c

c c

c

+ + =

+ =

=

3 2 90 180

5 90 180

5 90

y y

y

y

c c

c c

c

+ + =

+ =

=

w w

w

w

30 180

2 30 180

2 150

c c

c c

c

+ + =

+ =

=

2 2 8 180

12 180

z z z

z

c

c

+ + =

=

1 a bX

W Z

Y Q

P S

R


13HSERIES TOPIC


` +KLM + + + = 360c

` +KLM + = 360c

` +KLM =

2

3

a

a

b

b

360ABC BCD ADC BAD c+ + + ++ + + = 360KLM LMN KNM LKN c+ + + ++ + + =

` + +BCD + + = 360c

` + +BCD = 360c

` +BCD =

E F

G

D

W

X

O

Y

A

D

B

C

K

L

N

M

110c

80c

113c

32c

120c

110c

105c

62c24c

49c

44c

242c


Page 7 questions

Page 8 questions

113c 44c

303c 196c

57c 164c

80c 32c110c 120c

DEF 83c+ = OYX 45c+ =


4 a b c

q = q = q =

q q

q

135c 45c

115c

77c 49c

41c

30c68c

90c 100c 240c

13HSERIES TOPIC



x = y = z =

r = s = t =

Page 8 questions

89c

60c 20c 18c

83.7c 21.1c


5

6

a

a

b

b

c

c

r

s

t

85c 112.5c

134c130c

2y

5y

7z

3z7z

3z38c

68c

2x

52.6c 110c49.8c

82.3c

113.7c73.5c

206.8c

x

x

x

x

2 38 68 134 360

2 240 360

2 120

60

c c c c

c c

c

c

+ + + =

+ =

=

=

y y

y

y

y

2 5 90 130 360

7 220 360

7 140

20

c c c

c c

c

c

+ + + =

+ =

=

=

z z z z

z

z

3 7 3 7 360

20 360

18

c

c

c

+ + + =

=

=


13HSERIES TOPIC


Internal angle sum

= # angle sum of a triangle

= #

=

Internal angle sum


= #

=

7

Split into triangles:

Quadrilateral:

D & N F & GC & I H & ME & K A & L B & J

1

1

1

1

1

1

2

2

2

22

2

5

55

3

3

333

3

6

6

6

4

4

4

4

4

7

91c

97c

116c

78c

1 2

92c

5

3

110c6

4

58c

253c7

D CE

K

G

FI

N

MH A

L

J

B


Page 9 questions

Page 11 questions

1

2

a

a

b

b

c

Number of triangles = Number of triangles = Number of triangles =

Angle sum of a polygon

4

3 4

540c 720c

3 4180c 180c

6 7

7

13HSERIES TOPIC



Internal angle sum =

Each internal angle = '

=

Page 11 questions

Page 12 questions

2 c d

Internal angle sum


= #

=

Internal angle sum


= #

=



8 6

1440c 1080c

8 6180c 180c

3 a b


Each internal angle = '

= to 1 d.p.

720c 900c

720c 900c

144c 128.6c

5 7


13HSERIES TOPIC


Calculate the internal angle sum of these polygons and then calculate the size of the missing angle.4

a

c

b

Internal angle sum = Internal angle sum =

` x =


` y =

` z =

` x = 540c - (115c + 130c + 85c + 112c)

540c

115c y

112c

218c

98c

67c

x

112c 85c

130c

3z

120c

132c

140c 156c

2z

135c

142c


Page 12 questions

720c

1080c

51c

135c98c

720 90 218 98 67 112y` c c c c c c= - + + + +^ h

5 1080 142 135 156 140 132 120

5 255

z

z

` c c c c c c c

c

= - + + + + +

=

^ h

13HSERIES TOPIC


Where does it work? Solutions Angles and Polygons

Each angle = [( - ) # 180c] '

= [ # 180c] '

= '

=

n =

Each angle = [( - ) # 180c] '

= [ # 180c] '

= '

=

8

8

6

1080c

135c 156c

8

8

2 8

Page 14 questions

Internal angle rule for polygons

1

2

3

4

a

a

a

a b

b

b

b

Angle sum of a decagon = 10 2 180# c-^ h

= 8 180# c

= 1440c

Angle sum of a dodecagon = 18012 2 # c-^ h

= 18010 # c

= 1800c

Each internal angle of an icosagon Each internal angle of a heptagon

n = n =

n =

Angle sum = ( - ) # 180c

= # 180c

=

Angle sum = ( - ) # 180c

= # 180c

=

6 9

6 9

4 7

720c 1260c

2 2

15

15

13

2340c 15

15

2 15

20 2 180 20

3240 20

162

# '

'

c

c

c

= -

=

=

^ h6 @

(to d.p.)147.3 1

11 2 180 11

1620 11

# '

'

c

c

c

= -

=

=

^ h6 @


13HSERIES TOPIC


n = ( ' 180c) + 2

= sides

n = 7

n = 11

720c 2520c

166

n =

n =

10

17

5

6

a

a

c

c

b

b

b

d

d


n = ( ' 180c) + 2

= sides

Page 15 questions

Internal angle sum = 7 2 180# c-^ h

= 5 180# c

= 900c

Internal angle sum = 18011 2 # c-^ h

= 1809 # c

= 1620c


= 1808 # c

= 1440c


= 18015 # c

= 2700c

sides (dodecagon)

2

10 2

12

n 1800 180'c c= +

= +

=

^ h

sides (icosikaidigon)

n 3600 180 2

20 2

22

'c c= +

= +

=

^ h

13HSERIES TOPIC




Page 16 questions

1 2 3 4 5 6 7 8 9

Start anywhere along here

Finish anywhere along here

L RA IB NY T H

710c

6 = W

720c

9 = R720c

8 = V1080c

3 = D

900c

3 = D

540c

1 = T

1080c

2 = I

1260c

6 = F

1260c

4 = Y

1080c

1 = B

540c

6 = I540c

4 = A570c

2 = Z540c

3 = J450c

1 = D530c

6 = M540c

7 = K520c

2 = Z520c

6 = P540c

5 = Y

910c

7 = M

1260c

5 = H

1800c

8 = C720c

9 = S

720c

9 = H

1440c

4 = M

1980c

8 = T720c

5 = R

540c

1 = L

180c

1 =T

700c

5 = S

1440c

2 = A

1008c

2 = W

980c

7 = H

180c

9 = R

1080c

3 = T

920c

3 = T

900c

9 = L

1400c

4 = A

1800c

1 = S

900c

7 = N

720c

9 = L

900c

3 = T

710c

8 = E 690c

1 =A

1240c

8 = R

520c

7 = Y900c

2 = U

910c

4 = C

180c

9 = I

620c

4 = N720c

3 = B

700c

7 = D710c

4 = P

720c

2 = G720c

4 = E720c

5 = O


13HSERIES TOPIC


External angle rule for polygons

1

2

3

a

a

a

b

b

b

c

c

c

is produced to is produced to is produced to

W

X

YZ

D

R

Q

QS DC MNT G Q

S

T

A

E

F C

G L

K

P

O

N

Q

M

B

D

B

A C

J

K

N

I M

L

V

125c70c

80c 75c

65c

z

z

y

135c

x

Page 18 questions

135 125 360

360 260

100

x

x

x

c c c

c c

c

+ + =

= -

=

90 90 70 360

360 250

110

y

y

y

c c c c

c c

c

+ + + =

= -

=

80 65 75 360

2 360 220

2 140

70

z z

z

z

z

c c c c

c c

c

c

+ + + + =

= -

=

=

13HSERIES TOPIC



4

5

a

c

e

a

b

b

d

f

Each external angle =






As the number of sides increases, the size of each external angle increases / decreases.

As the number of sides increases, the size of each internal angle increases / decreases.


Page 19 questions

120c

72c

51.4c

90c

60c

45c


13HSERIES TOPIC


6

7

a

a

b

b c

a =

b =

c =

d =

e =

f =

100c 65c 79.5c

72c 69c 125c

108c

a

b 115c c d 111c

f

e

55c

100.5c

114c 128c k

lj120c

35c2x

82c

2x35c

60c

60c y

y

120 180

60

j

j

c c

c

+ =

=

128 180

52

k

k

c c

c

+ =

=

60 114 128 360

360 302

58

l

l

l

c c c c

c c

c

+ + + =

= -

=

60 60 35 35 82 2 2 360

272 4 360

4 88

22

x x

x

x

x

c c c c c c

c c

c

c

+ + + + + + =

+ =

=

=

2 180

180 44

136

x y

y

y

c

c c

c

+ =

= -

=


Page 20 questions

13HSERIES TOPIC




Page 21 questions

8

9

(i)

(ii)

2a

110c

y

x

130c

74c

52c

3c

55c

b

a

2b

80c

A

BC

Q

D

P

E

Angle sum 5 2 180

3 180

540

#

#

c

c

c

= -

=

=

^ h

80 180

100

x

x

c c

c

+ =

=

100 110 130 90 540

540 430

110

110 180

70

BCD

BCD

BCD

y

y

c c c c c

c c

c

c c

c

+

+

+

+ + + + =

= -

=

+ =

=

2 74 180

2 180 74

2 106

53

a

a

a

a

c c

c c

c

c

+ =

= -

=

=

2 180

3 180

60

b b

b

b

c

c

c

+ =

=

=

52 74 53 60 55 3 360

3 360 294

3 66

22

c

c

c

c

c c c c c c

c c

c

c

+ + + + + =

= -

=

=


13HSERIES TOPIC


(i)

(ii)

(iii)

10


Page 22 questions

For regular polygons, each external angle size = 360c ' the number of sides

` The number of sides = 360c ' the external angle size

` There are seven regular polygons with external angles ranging from 20c through to 30c.

The number of sides

sides

360 20

18

'c c=

=

The number of sides

sides

360 30

12

'c c=

=

Trying external angles of 20c,

Trying external angles of 30c,

For regular polygons, each external angle side = 360c ' the number of sides

12 sides external angle size = 360c ' 12 = 30c

13 sides external angle size = 360c ' 13 = 27.7c to 1 d.p.

14 sides external angle size = 360c ' 14 = 25.7c to 1 d.p.


16 sides external angle size = 360c ' 16 = 22.5c

17 sides external angle size = 360c ' 17 = 21.2c


The number of sides = 360c ' the external angle size

= 360c ' 50c

= 7.2 sides

Since there are not an exact whole number of sides, there cannot be a regular polygon with an external angle size of 50c

Therefore, Tara was correct.

13HSERIES TOPIC


What else can you do? Solutions Angles and Polygons

Page 24 questions

External angle of a triangle

1

2

3

a

a

a

b

b

b

c

c

a = +

=

x = -

=

b = +

=

y = -

=

c = +

=

z = -

=

45c a

x

42c

80c 71c y

z95c

44c

18c

50c

120c 30c25c

bc

48c 43c

89c

q

p

65c

35c r

s

45c

80c 95c 44c42c 71c 18c

38c 24c 26c

120c 90c

95c 150c 115c

50c 30c 25c

89 43

132

p

p

c c

c

= +

=

65 35

30

r

r

c c

c

= -

=

48 43

91

q

q

c c

c

= +

=

180

180 30

150

s r

s

s

c

c c

c

= -

= -

=


13HSERIES TOPIC


4 a

c

e

b

d

f100c

4x

b5a

2a

a

50c

j j k


160c5n

3n

q p

p

p

5m

l

Page 25 questions

180 50 2j

j 65

'c c

c

= -

=

^ h 180 3

60

p

p

'c

c

=

=

q

q

2 60

120

# c

c

=

=

65 50

115

k

k

c c

c

= +

=

180 90 2

45

l

l

'c c

c

= -

=

^ h

160 8

n n

n

n

n

5 3 160

8 160

20

'

c

c

c

c

+ =

=

=

=

5 90 45

5 135

27

m

m

m

c c

c

c

= +

=

=

4 4 100

8 100

12.5

x x

x

x

c

c

c

+ =

=

=

2 5 180

8 180

22.5

a a a

a

a

c

c

c

+ + =

=

=

22.5 45

67.5

b

b

c c

c

= +

=

13HSERIES TOPIC




Page 25 questions

5

(iii)

(i) and (ii)C

FA E

B D

3y

4z

4x 4x

4x

4x 5x

5x

5x 5x

6x

6x 6x

6x

C

FA E

BD

There are MANY combinations. Here are a four:

ABF BFA FAB

ACE CEA CAE

ABF FBD BDF

BDC DFE DEF

180

180

180

180

c

c

c

c

+ + +

+ + +

+ + +

+ + +

+ + =

+ + =

+ + =

+ + =

4 5 6 180

15 180

180 15

12

x x x

x

x

x

'

c

c

c

c

+ + =

=

=

=

, ,4 48 5 60 6 72x x xc c c= = =

3 6 4

3 72 48

3 120

40

y x x

y

y

y

c c

c

c

= +

= +

=

=

z x x

z

z

z

4 6 5

4 72 60

4 132

33

c c

c

c

= +

= +

=

=


13HSERIES TOPIC

Notes Angles and Polygons

www.mathletics.com

Angles and Polygons

angles and polygons - navigate mathnavigatemath.weebly.com/uploads/7/...anglespolygons... · 2...

Documents