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Angles andPolygons
Solutions
13HSERIES TOPIC
1Angles and Polygons SolutionsMathletics Passport © 3P Learning
How does it work? Solutions Angles and Polygons
Page 3 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
How does it work? Solutions Angles and Polygons
Angle sum of a triangle
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
Page 4 questions
EGF 60c+ = UVW 98c+ = YXZ 41c+ =
47c
59.5c
54.4c
71c
33.5c
21.7c
119c
68.7c
88.6c
13HSERIES TOPIC
3Angles and Polygons SolutionsMathletics Passport © 3P Learning
How does it work? Solutions Angles and Polygons
+PQR =
+QRS =
+RSP =
+QPS =
Sum =
Angle sum of a quadrilateral
+WXY =
+XYZ =
+YZW =
+XWZ =
Sum =
x = y = z =
v = w = u =
Page 5 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
Angle sum of a triangle
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
4 Angles and Polygons SolutionsMathletics Passport © 3P Learning
13HSERIES TOPIC
How does it work? Solutions Angles and Polygons
` +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
Angle sum of a quadrilateral
Page 7 questions
Page 8 questions
113c 44c
303c 196c
57c 164c
80c 32c110c 120c
DEF 83c+ = OYX 45c+ =
Angle sum of a quadrilateral
4 a b c
q = q = q =
q q
q
135c 45c
115c
77c 49c
41c
30c68c
90c 100c 240c
13HSERIES TOPIC
5Angles and Polygons SolutionsMathletics Passport © 3P Learning
How does it work? Solutions Angles and Polygons
x = y = z =
r = s = t =
Page 8 questions
89c
60c 20c 18c
83.7c 21.1c
Angle sum of a quadrilateral
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
+ + + =
=
=
6 Angles and Polygons SolutionsMathletics Passport © 3P Learning
13HSERIES TOPIC
How does it work? Solutions Angles and Polygons
Internal angle sum
= # angle sum of a triangle
= #
=
Internal angle sum
= # angle sum of a triangle
= #
=
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
Angle sum of a quadrilateral
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
7Angles and Polygons SolutionsMathletics Passport © 3P Learning
How does it work? Solutions Angles and Polygons
Internal angle sum =
Each internal angle = '
=
Page 11 questions
Page 12 questions
2 c d
Internal angle sum
= # angle sum of a triangle
= #
=
Internal angle sum
= # angle sum of a triangle
= #
=
Angle sum of a polygon
Angle sum of a polygon
8 6
1440c 1080c
8 6180c 180c
3 a b
Internal angle sum =
Each internal angle = '
= to 1 d.p.
720c 900c
720c 900c
144c 128.6c
5 7
8 Angles and Polygons SolutionsMathletics Passport © 3P Learning
13HSERIES TOPIC
How does it work? Solutions Angles and Polygons
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 =
Internal angle sum =
` 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
Angle sum of a quadrilateral
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
9Angles and Polygons SolutionsMathletics Passport © 3P Learning
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 @
10 Angles and Polygons SolutionsMathletics Passport © 3P Learning
13HSERIES TOPIC
Where does it work? Solutions Angles and Polygons
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
Internal angle rule for polygons
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
Internal angle sum = 18010 2 # c-^ h
= 1808 # c
= 1440c
Internal angle sum = 18017 2 # c-^ h
= 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
11Angles and Polygons SolutionsMathletics Passport © 3P Learning
Where does it work? Solutions Angles and Polygons
Internal angle rule for polygons
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
12 Angles and Polygons SolutionsMathletics Passport © 3P Learning
13HSERIES TOPIC
Where does it work? Solutions Angles and Polygons
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
13Angles and Polygons SolutionsMathletics Passport © 3P Learning
Where does it work? Solutions Angles and Polygons
4
5
a
c
e
a
b
b
d
f
Each external angle =
Each external angle =
Each external angle =
Each external angle =
Each external angle =
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.
External angle rule for polygons
Page 19 questions
120c
72c
51.4c
90c
60c
45c
14 Angles and Polygons SolutionsMathletics Passport © 3P Learning
13HSERIES TOPIC
Where does it work? Solutions Angles and Polygons
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
+ =
= -
=
External angle rule for polygons
Page 20 questions
13HSERIES TOPIC
15Angles and Polygons SolutionsMathletics Passport © 3P Learning
Where does it work? Solutions Angles and Polygons
External angle rule for polygons
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
+ + + + + =
= -
=
=
16 Angles and Polygons SolutionsMathletics Passport © 3P Learning
13HSERIES TOPIC
Where does it work? Solutions Angles and Polygons
(i)
(ii)
(iii)
10
External angle rule for polygons
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.
15 sides external angle size = 360c ' 15 = 24c
16 sides external angle size = 360c ' 16 = 22.5c
17 sides external angle size = 360c ' 17 = 21.2c
18 sides external angle size = 360c ' 18 = 20c
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
17Angles and Polygons SolutionsMathletics Passport © 3P Learning
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
= -
= -
=
18 Angles and Polygons SolutionsMathletics Passport © 3P Learning
13HSERIES TOPIC
What else can you do? Solutions Angles and Polygons
4 a
c
e
b
d
f100c
4x
b5a
2a
a
50c
j j k
External angle of a triangle
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
19Angles and Polygons SolutionsMathletics Passport © 3P Learning
What else can you do? Solutions Angles and Polygons
External angle of a triangle
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
= +
= +
=
=
20 Angles and Polygons SolutionsMathletics Passport © 3P Learning
13HSERIES TOPIC
Notes Angles and Polygons
www.mathletics.com
Angles and Polygons