Download Calculating angles - Pearson Schools and FE Colleges

02 Section 2 pp014-025.qxd
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Key words
Calculating angles
Know and use the fact that the sum of the angles in a triangle is 180°
Know and use the fact that the sum of the angles in a quadrilateral is 360°
Understand that the exterior angle of a triangle is equal to the sum of the
two interior opposite angles
c
a
b
Alternate
angles
proof
interior
exterior
Alternate
angles
c
a
p
q
r
s
a b c 180° (angles on a straight line)
All quadrilaterals can be
split into two triangles.
p q r s 360°
The sum of the angles in any triangle
is 180°.
The sum of the angles in any
quadrilateral is 2 180 360°.
An interior angle is inside the shape.
An exterior angle is outside the shape.
It is made by extending one of the lines.
c e 180°
(angles on a straight line)
a b c 180°
(angles in a triangle)
so a b e
b
e
c
a
The exterior angle of a triangle is equal to the sum of the two interior opposite angles.
Example
Calculate the size of the lettered angles, giving reasons for your answers.
a)
b)
2x
65°
c
a) c 65° 140°
140°
60°
b) 2x x 60° 180°
c 140° 65°
75°
3x 60° 180°
3x 120°
x 40° and 2x 80°
The exterior angle of a triangle is equal to the
sum of the two interior opposite angles.
14
Maths Connect 2R
x
The sum of
angles in a
triangle is 180°.
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Exercise 2.1 .............................................................................................
Calculate the size of the lettered angles, giving reasons for your answers.
a)
150°
b)
B
d
c)
b
C
30°
130°
a
29°
A
d)
e)
114°
34°
h
D
i
23°
104°
57°
Work out the value of x in each of the following:
a)
b)
c)
2x
3x
x
64°
x
40°
2x
50°
100°
x
Calculate the size of the lettered angles, giving reasons for your answers. Make sure that
you show each step of your working.
a)
c
b)
40°
c)
60°
2x
78°
x
z
5x
y
2x
Make up a question that involves using angle facts to find the size
Look at the questions
above to help you.
of lettered angles. Ask your partner to calculate the size of
the lettered angles.
Investigation
Draw two straight lines AB and CD.
Mark a point O between them.
Draw a line through O that crosses the two lines.
Mark the two angles underneath the line as x and y.
Measure the size of x and y with a protractor.
Draw some more lines through O. Each time
measure the angles underneath the line and record
your results in a table. What do you notice?
Can you explain why this happens?
Can you prove what you have found?
C
A
O
O
x
B
C
A
D
B
y
D
Extend AB and CD so they meet.
Calculating angles 15
02 Section 2 pp014-025.qxd
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Key words
Angle sums of polygons
Explain how to find, calculate and use the sums of the interior and
exterior angles of quadrilaterals, pentagons and hexagons
Solve problems using properties of angles, of parallel and intersecting
lines, and of triangles and other polygons
interior angles
pentagon
angle sum
hexagon
polygon
In order to calculate the sum of the interior angles of a
180°
pentagon , we can split it into triangles:
Each triangle has an angle sum of 180°. There are three triangles
so the total angle sum is 3 180° 540°. This is true of any
180°
180°
pentagon so the interior angle sum of a pentagon is 540°.
In order to calculate the sum of the interior angles of a hexagon ,
we can split it into triangles:
Each triangle has an angle sum of 180°. There are four
triangles so the total angle sum is 4 180° 720°. This is true
of any hexagon so the interior angle sum of a hexagon is 720°.
A polygon with n sides can be split into n 2 triangles.
Each triangle has an angle sum of 180° so the interior angle
sum of any polygon is (n 2) 180°.
Example
180°
180°
180°
There are always two
triangles less than the
number of sides of the
polygon.
Calculate the size of the lettered angles, giving reasons for your working.
a)
b)
x
72°
n
58°
l
120°
m
y
174°
147°
a) x 180 72 (Angles on a straight line add up to 180°.)
108°
y 540 108 120 90 90
132° (The sum of the angles of a pentagon is 540°.)
b) n 180 58 (Angles on a straight line add up to 180°.)
122°
m 58° (It is a corresponding angle to the angle of 58°.)
l 720 122 58 90 147 174
129° (The sum of the angles of a hexagon is 720°.)
16
180°
Maths Connect 2R
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Exercise 2.2 .............................................................................................
Calculate the size of the lettered angles, giving reasons for your working.
a)
b)
B
d
147°
c)
C
Q
48°
b
A 120°
89° D
P
c
60°
a
152°
a
F
120°
E
76°
R
b
72°
S
T
Calculate the size of the following angles, giving reasons
for your working.
a) AED
b) ADE
d) BCD
e) ABC.
A
48°
c) ADC
B
50°
115°
E
C
D
Calculate the value of y in each of these diagrams, showing all your working.
a)
b)
130°
y
110°
y
118°
130°
y
reasons for your working.
45°
2y
100°
Calculate the value of x, giving
c)
2y
y
y
a)
b)
x
70°
30°
x
a) Draw any pentagon and measure the size of the interior angles.
b)
c)
d)
e)
Mark them on your diagram. Calculate the size of each of the
exterior angles and add them together.
Write down your total.
Draw any hexagon and measure the size of the interior angles.
Mark them on your diagram. Calculate the size of each of the
exterior angles and add them together.
Write down your total.
What do you notice about the total sum of the exterior angles
of the pentagon and hexagon?
Is the same true for other polygons?
Try triangles and
quadrilaterals first.
Can you explain why this happens?
Can you give a proof?
Angle sums of polygons 17
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Key words
Angle sums of regular polygons
regular
interior angle
exterior angle
vertex
Calculate and use the interior and exterior angles of regular polygons
Solve problems using properties of angles
A regular shape has all sides of equal length and all angles equal.
Look at this regular pentagon.
The interior angles are all 108° and the
exterior angles are all 72°.
Imagine the pentagon as a path. At each vertex there
is a change of direction.
By walking around the complete polygon so that you
finish by facing in the same direction as you started, a
whole turn of 360° will be made.
The exterior angles sum to 360°.
We can prove that this is true of any polygon.
If the exterior angle sum is 360°, then each exterior
360°
angle of a regular polygon is .
n
72°
108°
72°
108°
108°
72°
72° 108°
108°
72°
72 72 72 72 72 360°
For a regular pentagon this
can be described using LOGO.
Input ‘repeat 5 [Fd20 Rt72].’
For a regular pentagon,
360
this is 72°.
5
Since the interior and exterior angles sum to 180°,
each interior angle of a regular polygon is
360
360°
For a regular pentagon, this is 180 108°.
180° .
5
n
Example 1
Calculate the sizes of the exterior and interior angles of a regular 20-sided
polygon.
360
360
Each exterior angle of a regular polygon n
20
Substitute 20 into the
formula.
18°
Each interior angle 180 18 162°.
Example 2
A regular polygon has interior angles of 135°. How many sides does it have?
What is the name of the regular polygon?
The exterior angle 180° 135 45°
360
Each exterior angle of a regular polygon , where n is the
n
number of sides.
360
If 45, then 45n 360 and n 360 45 8.
n
The polygon has eight sides. It is a regular octagon.
18
The interior and exterior
angles sum to 180°.
Maths Connect 2R
45°
135°
45°
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Exercise 2.3 .............................................................................................
Calculate the size of the exterior and interior angles of regular polygons with the
following numbers of sides:
a) 6
b) 10
e) 120
f) 24
c) 12
g) 36
d) 90
h) 60
Regular polygons have interior angles of the following sizes:
a) 140°
b) 175°
c) 150°
For each polygon, find the number of sides.
Look at Example 2.
d) 168°
ABCDE is a regular pentagon.
Copy the diagram and mark the sizes
of all the coloured angles.
ABCDEFGH is a regular octagon.
Work out the size of:
a) BCA
b) DAC.
B
C
A
D
H
E
G
F
In a regular polygon, each interior angle is 120° greater than each exterior angle.
Calculate the number of sides that the polygon has.
Look at the parallelogram ABCD.
B
C
Can you prove why ABC and DAB
sum to 180°?
Try extending the line AB.
A
D
Investigation
a) Two sides of a regular pentagon are extended to form a
x
triangle with angle x.
Calculate the value of x, showing all the steps of your
working and giving reasons for your answer.
Prove that the triangle is isosceles.
b) Two sides of a regular ten-sided polygon are extended to form
the angle y.
Calculate the value of y showing all the steps of your working
and giving reasons for your answer.
c) Investigate the angles made by extending two sides of another
regular polygon.
y
Angle sums of regular polygons 19
02 Section 2 pp014-025.qxd
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Key words
Construction
bisector
compasses
equidistant
perpendicular
bisector
mid-point
Construct a bisector of an angle, using a ruler and a pair of compasses
Construct the mid-point and perpendicular bisector of a line segment,
using a ruler and a pair of compasses
The bisector of an angle is a line that divides the angle
into two equal parts. You can construct the bisector of an
angle using a pair of compasses .
A
x
x
B
D
In this diagram, BD is the bisector of the angle ABC.
Every point on the line BD is equidistant from the
lines BA and BC.
C
B
The perpendicular bisector of a line segment
divides the line segment into two equal parts at
right angles.
M
A
In this diagram, BD is the perpendicular bisector of
AC. It crossses AC at the mid-point (M) of the line.
Every point on the line BD is equidistant from
both A and C. If you join ABCD, where BM DM, a
rhombus is formed.
Example 1
D
ABC is 60°. Construct the bisector of ABC.
A
A
C
B
C
A
C
B
Open the compasses and put
the point on B. Draw an arc
that intersects with BA and BC.
D
C
B
Do not change the opening of the compasses. Put the point first on the intersection
of the arc with BA and then with BC. Draw new arcs to intersect at D. Join BD.
Example 2
The line segment AB is 2.5 cm long. Construct
the perpendicular bisector of AB and label it CD.
Join AC, BC, BD and AD to form a rhombus.
Draw a line AB of 2.5 cm.
Open up the compasses to
over half this length.
C
A
2.5 cm
B
A
B
A
B
D
20
Maths Connect 2R
Place the point at A and
draw an arc. Keep the
opening of the compasses
the same and repeat at B.
Label the points where the
arcs intersect C and D. Join
CD, then AC, BC, BD and AD.
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Exercise 2.4 .............................................................................................
a) Draw any acute angle.
Construct the angle bisector using only a ruler and a pair of compasses.
b) Draw any obtuse angle.
Construct the angle bisector using only a ruler and a pair of compasses.
Draw a line segment AB that is 10 cm in length.
Construct the perpendicular bisector of AB using only a ruler and a pair of compasses and
label it CD. Join AC, BC, BD and AD. What is the resulting shape? Justify your answer.
This question can be done on paper or using ICT.
Draw some lines for your partner. Ask them to construct the perpendicular bisectors of
the lines and to mark the mid-points. Next draw some angles for your partner to bisect. If
using paper, measure your partner’s accuracy.
Draw any reflex angle.
Construct the angle bisector using only a ruler and a pair of compasses.
Check it is correct by measuring the two angles and making sure they are the same.
A new water channel is to be built at the zoo.
Animal
house 1
It is to be placed between two animal houses so
that the houses are equidistant from the channel.
Copy this diagram, and construct a red line to show
the position of the water channnel.
Animal
house 2
There is one piece of Emily’s birthday cake left.
Emily and Rajeev decide to divide the piece of cake into two
equal pieces so that they can share it between them.
Copy this diagram of the piece of cake and construct a red line
to show where Emily should slice it.
Investigations
Draw any triangle with sides longer than 4 cm but shorter than 8 cm. Construct the
angle bisector for each of the three angles. Make sure that the angle bisectors are
long enough to cross each other. What do you notice?
B
The diagonal AC of a rhombus ABCD is 4 cm long.
Investigate possible lengths for the other diagonal, BD.
What happens when AC BD?
A
C
Here is one example.
D
Construction 21
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Key words
Perpendiculars
construct
perpendicular
arc
Use a straight edge and pair of compasses to construct the
perpendicular from a point to a line and from a point on a line
We can use a ruler and pair of compasses to:
P
● construct the perpendicular from a point to a line
N
● construct the perpendicular from a point on a line
P
An arc is part of a circle. You can draw arcs with your compasses.
Example 1
Make a copy of the diagram.
Using only a ruler and pair of compasses draw
a perpendicular from the point P on the line.
1)
P
2)
Q
Q
P
P
1) Using compasses, draw two arcs with the same radius from P to make two intersections on the line. From
the two intersections, draw two arcs with the same radius that intersect and label the intersection Q.
2) Place a straight edge from P to Q and join them with a straight line.
Example 2
1)
Make a copy of the diagram.
Using only a ruler and pair of compasses draw
a perpendicular from the point A to the line.
2)
A
B
A
A
B
1) Using compasses, draw two arcs with the same radius from the point A that intersect with the line. From
the two arcs draw two more arcs with the same radius that intersect and label the intersection B.
2) Place a straight edge from A to B and join A to the straight line.
22
Maths Connect 2R
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Exercise 2.5 .............................................................................................
Make a copy of the diagram. Using a ruler and pair of
compasses, construct a perpendicular from the point D
on the line.
2 cm
D
3.5 cm
Make a copy of the diagram. Using a ruler and pair of compasses,
construct a perpendicular from the point E to the line.
E
Make a copy of the diagram. Using a ruler and a pair of
Y
compasses, construct a perpendicular from the point X on
the line and a perpendicular from point Y to the line.
Jamina is walking her dog in the park
when it starts to rain. She wants to take
the shortest route back to the path.
Copy the diagram and construct the
shortest route to the path, using a ruler
and a pair of compasses.
Reuben is at the beach and wants to swim the shortest
X
Jamina
Path
Raft
route to a raft fixed to the seabed.
Copy the diagram and construct the shortest route to
the raft, using a ruler and a pair of compasses.
Beach
Investigation
a) Draw any triangle. Construct the perpendiculars from each vertex to the
opposite side. What do you notice?
b) Draw another triangle. Construct the perpendicular bisectors of each side.
Measure the distance from the intersection of the bisectors to each vertex. What
do you notice? Draw a circle centred on the intersection through each vertex.
Draw more triangles to check that it always works. Find out the name for this
point.
Perpendiculars 23
02 Section 2 pp014-025.qxd
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Page 24
Key words
Constructing right-angled triangles
hypotenuse
Use a straight edge and a pair of compasses to construct a triangle,
given a right angle, the hypotenuse and a side (RHS)
Previously you have constructed unique triangles given:
S
S
S
A
A
A
S
S
S
two sides and the
included angle (SAS)
two angles and the side
between them (ASA)
three sides (SSS)
If we are given this information, we can be sure that the triangle we are drawing is the
only one of its kind.
In lesson 2.5 we practised constructing the perpendicular from
a point on a line.
We can use this knowledge to construct a right-angled
triangle, using a ruler and a pair of compasses, if we
Right
know the length of the hypotenuse and the length of
angle
one of the sides (RHS).
The hypotenuse is the longest side
of a right-angled triangle.
Example
Hypotenuse
Side
Construct a triangle ABC with a right angle, a hypotenuse BC of length 5 cm
and a side AC of length 3 cm. Measure and write down the size of the other
two angles, ABC and BCA.
B
37°
4 cm
5 cm
53°
A
A
3 cm
C
A
3 cm
C
ABC 37° and BCA 53°.
First, draw a line of more than
3 cm in length. Mark a point A.
Construct the perpendicular
from A and draw in the right
angle.
24
Maths Connect 2R
Measure a horizontal distance
of 3 cm from A and mark in the
point C.
Open your compasses out to 5 cm
and place the point on C. Draw an
arc that intersects with the line
perpendicular to AC. Mark this
point B. Complete the triangle and
measure ABC and BCA.
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Exercise 2.6 .............................................................................................
Use plain paper for the constructions in this exercise.
Construct the following right-angled triangles. Measure and write down the length of the
third side and the size of the other two angles.
a) hypotenuse 8 cm, side 5 cm
b) hypotenuse 6 cm, side 3 cm.
Construct these right-angled triangles to scale. Measure and write
down the length of the third side and the size of the other two angles.
a)
b)
30 cm
15 cm
Use a scale of
‘1 cm represents
10 cm’.
c)
75 cm
15 cm
120 cm
50 cm
The diagram shows the wooden frame of a roof.
8m
The vertical height of the roof is 3 m and the length of the
slope is 8 m.
a) Use this information to construct a scale drawing of the
outline of the roof.
b) Use your drawing to find:
i) the angle the roof makes with the horizontal, ii) the actual width of the roof.
3m
A ladder that is 5 m long leans against a wall with its base 2 m from the wall.
a) Use this information to construct a scale drawing
of the ladder and the wall.
b) Use your drawing to find:
i) how far the ladder reaches up the wall
ii) the angle between the ladder and the ground.
Remember to find the actual
length, not the length on your
scale drawing.
A piece of wire that is 15 m long is stretched from the top of a 12 m mast to a fixing in the
ground.
a) Use this information to construct a scale drawing.
b) Use your drawing to find how far the fixing is from the foot of the mast.
Investigation
If we know the length of two sides of a triangle and the size
of an angle that is not included between the sides (SSA),
there are three possible triangles we can draw.
You might find it helpful to use
One of the triangles will be right-angled.
ICT for this investigation.
Discover what happens in the other two cases.
Constructing right-angled triangles 25