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Chapter
5
Angles and Parallel Lines
Angles and parallel lines provide BC’s Liard River Bridge with strength, stability, and visual appeal. The bridge was built in 1942.
5.1
Measuring, Drawing, and Estimating Angles
REVIEW: WORKING WITH ANGLES
In this section, you will review types of angles and how to classify them.
An angle is formed when two rays meet at a point called the vertex. Angles are usually
measured in degrees using a protractor. Angle measures range from 0° to 360°.
Angles are: •• acute, if their measure is between 0° and 90°;
•• right, if their measure is 90°; the two rays are perpendicular to each other;
•• obtuse, if their measure is between 90° and 180°;
•• straight, if their measure is 180°; and
•• reflex, if their measure is between 180° and 360°.
Chapter 5 Angles and Parallel Lines
Example 1
Identify the type of angle: acute, right, obtuse, straight, or reflex.
a)
b)
c)
d)
e)
SOL U T ION
a) This is a right angle.
b) This is a straight angle.
c) This is an obtuse angle.
d) This is an acute angle.
e) This is a reflex angle.
BUILD YOUR SKILLS
1. Identify the type of angle: acute, right, obtuse, straight, or reflex.
a) 68° b) 215° c) 91° d) 32°
e) 180° f) 99° g) 195° h) 265°
215
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MathWorks 10 Workbook
NEW SKILLS: WORKING WITH REFERENT ANGLES
In many jobs, people have to draw angles or estimate their measure. To estimate the size
of an angle, you can use referent angles, which are angles that are easy to visualize. You
can use these referents to determine the approximate size of a given angle.
90°
45°
60°
30°
For more details, see page 174 of MathWorks 10.
Example 2
Use the referents above to estimate the size of each of the following angles. Use a
protractor to check your answers.
B
C
A
SOL U T ION
∠ is a symbol used
to indicate an angle.
∠A is slightly bigger than the 45° referent. It is probably about 50°.
∠B is less than 30°, so it is probably between 15° and 20°.
∠C is close to a straight angle but it is probably about 10° less. Therefore, ∠C is
approximately 170°.
Using a protractor, measure the angles.
∠A is 52°.
∠B is 18°.
∠C is 172°.
Chapter 5 Angles and Parallel Lines
BUILD YOUR SKILLS
2. Use the referents to determine the approximate size of the following angles.
A
B
C
D
3. What is the approximate angle of the railing on the stairs?
4. Use the referents in the New Skills section to determine the approximate size of the
following angles.
B
A
C
D
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MathWorks 10 Workbook
5. Jason is doing a survey of a city block. What is the approximate angle between his
sightings of the two buildings?
Example 3
complementary angles:
Given each of the following angles, determine the size of the complement and/or the
two angles that have
size of the supplement (if they exist).
measures that add up
to 90°
a) 75°
supplementary angles:
b) 43°
two angles that have
c) 103°
measures that add up
to 180°
d) 87°
e) 300°
Two angles with
the same measure
are referred to as
congruent.
SOL U T ION
To find the complement, subtract the angle measure from 90°.
To find the supplement, subtract the angle measure from 180°.
Chapter 5 Angles and Parallel Lines
a) Complement:
90° − 75° = 15°
Supplement:
180° − 75° = 105°
b) Complement:
90° − 43° = 47°
Supplement:
180° − 43° = 137°
c) Complement: The angle is already greater than 90° so there is no complement.
Supplement:
180° − 103° = 77°
d) Complement:
90° − 87° = 3°
Supplement:
180° − 87° = 93°
e) The angle is greater than 180°, so it has no complement or supplement.
BUILD YOUR SKILLS
6. Fill in the chart with the complement and the supplement of each angle, if they
exist. If they don’t exist, state why.
ANGLE COMPLEMENTS AND SUPPLEMENTS
Angle
45°
78°
112°
160°
220°
Complement
Supplement
219
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MathWorks 10 Workbook
7. The complement of an angle is 58°.
a) What is the size of the angle?
b) What is the supplement of the angle?
8. The complement of an angle is 0°.
a) What is the size of the angle?
b) What is the size of the supplement of the angle?
Chapter 5 Angles and Parallel Lines
221
NEW SKILLS: WORKING WITH TRUE BEARING
In navigation and map-making, people often measure angles from the vertical, or north.
true bearing: the angle
The angle, measured in a clockwise direction from a line pointing north, is referred to
measured clockwise
as the true bearing. Straight north has a bearing of 0°.
between true north and
an intended path or
For more details, see page 182 of MathWorks 10.
direction, expressed in
N
NNW
degrees
NNE
NW
NE
WNW
ENE
W
E
WSW
ESE
SW
SE
SSW
SSE
S
Example 4
A boat is heading directly southwest. What is its true bearing?
SOL U T ION
N
W
E
S
If the boat is heading southwest, measuring from the vertical will give you an
obtuse angle of 225° (45° beyond a straight angle).
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MathWorks 10 Workbook
BUILD YOUR SKILLS
9. If a boat is travelling 25° south of straight east, what is its true bearing?
10.What is the true bearing of a boat travelling south?
11. What is the true bearing of a boat travelling north-northwest?
PRACTISE YOUR NEW SKILLS
1. Identify the type of angle: acute, right, obtuse, straight, or reflex.
a) 56°
b) 91°
c) 270°
d) 170°
e) 43°
f) 192°
Chapter 5 Angles and Parallel Lines
2. Estimate, using referents, the size of the angles indicated in the diagrams.
a)
b)
x
x
c)
d)
y
x
x
3. If Renata cuts a rectangular tile diagonally, one of the acute angles formed is 65°.
What is the size of the other acute angle?
4. Pete is laying irregularly shaped paving stones. He needs to find one to fit in
position A. Approximately what size of angle will it have?
A
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224
MathWorks 10 Workbook
5. On the map below, what is the bearing from the following points?
N
Point A
Point B
Point C
a) A to B
b) B to C
Chapter 5 Angles and Parallel Lines
Angle Bisectors and Perpendicular Lines
225
5.2
NEW SKILLS: WORKING WITH ANGLE BISECTORS
To bisect something is to cut it into two equal parts. An angle is bisected by a ray that
angle bisector: a
divides it into two angles of equal measure. The ray that divides the angle is called an
segment, ray, or line that
angle bisector.
separates two halves of a
Perpendicular lines are two lines that form a right angle. A right angle (90°) can be
thought of as a bisected straight (180°) angle. The process used to draw perpendicular
lines is the same as drawing angle bisectors because a perpendicular line bisects a
straight angle.
For more details, see page 187 of MathWorks 10.
A
C
B
Point C bisects AB if AC is equal to CB.
A
B
D
Ray BD bisects ∠ABC if ∠ABD is equal to ∠DBC.
C
Example 1
Bisect ∠ABC using a straight edge and compass.
A
B
C
bisected angle
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MathWorks 10 Workbook
SOL U T ION
To draw a ray, BD, that bisects ∠ABC , follow these steps:
•• With the compass point at B, draw an arc to intersect BA and BC at X and Y,
respectively.
•• With the compass point at X and the radius more than half of XY, draw a small
arc in the interior of ∠ABC .
•• With the same radius and compass point at Y, draw a small arc to intersect
this arc at D.
•• Join B and D.
BD is the bisector of ∠ABC.
A
X
B
D
Y
C
BUILD YOUR SKILLS
1. If a right angle is bisected, what is the size of each angle?
2. Bisect the given angles using a straight edge and compass.
a)
b)
c)
Chapter 5 Angles and Parallel Lines
227
3. An angle is bisected. Each resulting angle is 78°. How big was the original angle?
4. The size of one resulting angle after the original angle is bisected is equal to the
supplement of the original angle. What is the measure of the original angle?
Example 2
Using a protractor, determine which of the following lines are perpendicular.
ℓ1
In the workplace,
carpenters often
ℓ2
use framing squares
and levels to ensure
that they have right
ℓ3
angles. A framing
square is a tool that
is a right angle.
ℓ4
ℓ5
SOL U T ION
The angles formed between ℓ1 and ℓ3 are each 90°, so ℓ1 and ℓ3 are perpendicular.
The angles formed between ℓ1 and ℓ4 are not 90°, so ℓ1 and ℓ4 are not perpendicular.
The angles formed between ℓ2 and ℓ3 are not 90°, so ℓ2 and ℓ3 are not perpendicular.
The angles formed between ℓ2 and ℓ4 are not 90°, so ℓ2 and ℓ4 are not perpendicular.
The angles formed between ℓ5 and ℓ3 are not 90°, so ℓ5 and ℓ3 are not perpendicular.
The angles formed between ℓ5 and ℓ4 are each 90°, so ℓ5 and ℓ4 are perpendicular.
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MathWorks 10 Workbook
BUILD YOUR SKILLS
5. A crooked table leg makes an angle of 86.7° with the tabletop. How much must the
carpenter move the leg so that it is perpendicular to the tabletop?
6. At what approximate angle does the hill incline from the horizontal?
x
7. A carpenter is inlaying different types of wood on a tabletop. What must be the size
of angles a, b, c, and d?
60°
a
75°
75°
b
55°
d
60°
75°
c
Chapter 5 Angles and Parallel Lines
PRACTISE YOUR NEW SKILLS
1. Use a protractor to determine whether these lines are perpendicular.
a)
c)
b)
d)
2. Complete the following table.
ANGLE CALCULATIONS
Angle
Complement
Supplement
Resulting angle
measure after the
angle is bisected
73°
12°
15°
132°
90°
34°
49°
68°
100°
127°
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230
MathWorks 10 Workbook
3. Kaleb is edging a garden bed with square tiles. In the corner shown below, he wants
two congruent tiles. At what angle must he cut the tiles so that they fit?
x
y
135°
4. Calculate the size of the indicated angles. Name as many pairs of complementary
and supplementary angles as possible.
a)
b)
144°
70°
y
x
x
c)
d)
115°
x 81°
x
5. The angle at the peak of a roof is 135°. Calculate the measure of the angle formed by
the rafter and the king post.
135°
king post
rafter