2 13.1 angle properties of parallel lines - mrvahora · 2 13.2 proving the angle properties of...
TRANSCRIPT
1 Find the missing angles in the following diagrams.
a ............................................... b ............................................... c ............................................... d ...............................................
2 Find all the missing angles in this diagram. Give reasons for your answers.
a .....................................................................................................................
b .....................................................................................................................
c .....................................................................................................................
d .....................................................................................................................
3 Find all the missing angles in this diagram. Give reasons for your answers.
a .....................................................................................................................
b .....................................................................................................................
c .....................................................................................................................
4 Prove that lines AB and CD are parallel.
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�2 13.1 Angle properties of parallel lines
159
a bd
135˚
c 72˚
a b
c d67˚
148˚A B
C D32˚
52˚
78˚
ab
c
Guided practice worksheet
D
Questions are targeted at the grades indicated
�2 13.2 Proving the angle properties of triangles and quadrilaterals
2161A
Write down a reason after each line of working, e.g. angle sum of a triangle.
1 Use the angle of a straight line (180°) to prove that opposite angles are equal.
a + b = ......................................
b + c = ......................................
So ...................................... = ......................................
So a = c
2 The base angles of an isosceles triangle are equal.
Prove that the bisector of the top angle meets the base at right angles.
You must show that c = 90°
a + b + c = ......................................
a + b + d = ......................................
So c = d
c + d = ......................................
So c = 90°
3 Prove that the diagonals of a square cross at right angles.
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Guided practice worksheet
b
dca
b
a c d a
b
ac
d
be
Hint Use isosceles triangles.
C
B
Questions are targeted at the grades indicated
�2 13.2 Proving the angle properties of triangles and quadrilaterals
3
Guided practice worksheet
161B
4 Prove that the sum of the angles of a pentagon is 540°.
You must show that a + b + c + d + e = 540°
a + t + p = ......................................
u + q + r = ......................................
s + d + v = ......................................
...................................... = ...................................... (add the equations together)
Replace and write p, q, r, s, t, u, v using angles b, c, e
p + q = ......................................
r + s = ......................................
t + u + v = ......................................
So a + b + c + d + e = 540°
5 Prove that the diagonals of a rhombus cross at right angles.
Write down the angles that are equal using isosceles triangles.
...................................... = ......................................
...................................... = ......................................
Write down the two angles that are equal using parallel lines.
...................................... = ......................................
Show that e = f
Show that e = 90°
qp
rs
dt u
v
a
b
c
de
a
bc
fe
a g
d
B
1 Work out the size of the missing angles in these quadrilaterals and triangles.
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.................................. ..................................
.................................. ..................................
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2 Explain why ACD cannot be a straight line.
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�2 13.3 Using the angle properties of triangles and quadrilaterals
163
80˚
75˚100˚
a
55˚ 70˚
123˚b
57˚
c
75˚
d
75˚
59˚
82˚
e f 35˚
h
g
115˚ 56˚
jik
80˚
75˚100˚
a
55˚ 70˚
123˚b
57˚
c
75˚
d
75˚
59˚
82˚
e f 35˚
h
g
115˚ 56˚
jik
80˚
75˚100˚
a
55˚ 70˚
123˚b
57˚
c
75˚
d
75˚
59˚
82˚
e f 35˚
h
g
115˚ 56˚
jik
80˚
75˚100˚
a
55˚ 70˚
123˚b
57˚
c
75˚
d
75˚
59˚
82˚
e f 35˚
h
g
115˚ 56˚
jik
80˚
75˚100˚
a
55˚ 70˚
123˚b
57˚
c
75˚
d
75˚
59˚
82˚
e f 35˚
h
g
115˚ 56˚
jik
80˚
75˚100˚
a
55˚ 70˚
123˚b
57˚
c
75˚
d
75˚
59˚
82˚
e f 35˚
h
g
115˚ 56˚
jik
80˚
75˚100˚
a
55˚ 70˚
123˚b
57˚
c
75˚
d
75˚
59˚
82˚
e f 35˚
h
g
115˚ 56˚
jik
80˚
75˚100˚
a
55˚ 70˚
123˚b
57˚
c
75˚
d
75˚
59˚
82˚
e f 35˚
h
g
115˚ 56˚
jik
50˚A
C
B
D
75˚
120˚
Guided practice worksheet Questions are targeted at the grades indicated
1 Measure the angle of elevation of the tree from the ground.
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2 Calculate the angle of depression from the telegraph pole to the ground.
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�2 13.4 Angles of elevation and depression
165
47˚
Guided practice worksheet Questions are targeted at the grades indicated
�2 13.5 Using angle properties to solve problems
167
1
a Calculate the size of angle a and hence prove that ACD is a straight line.
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b What type of triangle is ABC?
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2 a Calculate the size of angles a and b.
a = ..................................................................
b = ..................................................................
b What type of quadrilateral is ABCD?
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3 BEG and CFG are straight lines.
ABC is parallel to DEF.
Angle ABE = 56°
EF = EG
a Write down the size of the angle marked x.
..................................................................
b Give a reason for your answer.
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c Work out the size of the angle marked y. .................................................................. Give reasons for your answer.
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60˚
A
a
B
CD
85˚
95˚b
aA
B
C
56˚
x
y
B
F
G
DE
A C
Guided practice worksheet
D
C
Hint You will need to work out the angle GEF.
Questions are targeted at the grades indicated
�2 13.6 Polygons
169A
Use a protractor to measure the interior angles of these polygons. Fill your answers in the table.
Name of shape Size of angles Sum of interior angles
1
2
3
4
5
6
Guided practice worksheet
1
3
5
2
4
6
D
Questions are targeted at the grades indicated
�2 13.6 Polygons
169B
Guided practice worksheet
7 Complete the following
The sum of the interior angles for an n-sided polygon is (n – ......................) × 180°
8 Calculate the value of x.
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9 Calculate the sum of the interior angles for a decagon. ..................................................................
1 0 The sum of the interior angles for a regular polygon is 1980°. Write down the number sides of the polygon.
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1 1 A regular polygon has 15 sides.
a Calculate the size of its exterior angle. ..................................................................
b Calculate the size of its interior angle. ..................................................................
1 2 The diagram shows 2 sides of a regular polygon.
a Calculate the size of the interior angle of the polygon. ..................................................................
b Calculate the number of sides of the polygon. ..................................................................
36˚
D
C
85˚ 80˚
130˚
x
�2 13.6 Polygons
169C
1 3 AB and BC are two sides of a regular polygon. Explain why this polygon will not tessellate.
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Guided practice worksheet
45˚
B
C
A
C
�2 13.7–13.8 Isosceles triangle in a circle, tangents to a circle
171A
1 The diagrams all show circles, centre O.
Work out the size of each angle marked with a letter.
a b c
....................................... ....................................... .......................................
d e
.................................................................. ..................................................................
f g
.................................................................. ..................................................................
Guided practice worksheet
17° a
O O O132°
b33°
c
Remember that the two radii are equal so the triangles are isosceles triangles.
Questions are targeted at the grades indicated
O
42˚ d
O
36˚
e
O
52˚
fO
10˚
D
O
43˚
g
O
130˚F
O
42˚ d
O
36˚
e
O
52˚
fO
10˚
D
O
43˚
g
O
130˚F
O
42˚ d
O
36˚
e
O
52˚
fO
10˚
D
O
43˚
g
O
130˚F
O
42˚ d
O
36˚
e
O
52˚
fO
10˚
D
O
43˚
g
O
130˚F
D
B
�2 13.7–13.8 Isosceles triangle in a circle, tangents to a circle
171B
2 The diagrams all show circles, centre O.
Work out the size of each angle marked with a letter.
Give reasons for each answer.
a b
.................................................................. ..................................................................
c d
.................................................................. ..................................................................
e f
.................................................................. ..................................................................
O
40˚ k
UW
V
O
30˚
h
L
M
O33˚
i
N
P
Q
O
jR
T
S
O146˚
Xl
Y
O
10˚Z A
m
B
O
40˚ k
UW
V
O
30˚
h
L
M
O33˚
i
N
P
Q
O
jR
T
S
O146˚
Xl
Y
O
10˚Z A
m
B
O
40˚ k
UW
V
O
30˚
h
L
M
O33˚
i
N
P
Q
O
jR
T
S
O146˚
Xl
Y
O
10˚Z A
m
B
O
40˚ k
UW
V
O
30˚
h
L
M
O33˚
i
N
P
Q
O
jR
T
S
O146˚
Xl
Y
O
10˚Z A
m
B
O
40˚ k
UW
V
O
30˚
h
L
M
O33˚
i
N
P
Q
O
jR
T
S
O146˚
Xl
Y
O
10˚Z A
m
B
O
40˚ k
UW
V
O
30˚
h
L
M
O33˚
i
N
P
Q
O
jR
T
S
O146˚
Xl
Y
O
10˚Z A
m
B
Remember that you need to have a worded rule as a reason, not just a calculation.
Guided practice worksheet
B