8
Geometric
Relationships
The five-pointed star appears on the flag of more than 30 countries.
In Canada, the five-pointed star appears only on the flag of Nunavut.
1. What shapes do you see in the star?
Key Words
2. a) Which angles inside or outside the star are lesss
than 90 degrees?
reflex angle
interior angle
exterior angle
polygon
regular polygon
parallel lines
transversal
opposite angles
corresponding angles
co-interior angles
alternate angles
354 MHR • Chapter 8
b) Estimate the measure of these angles.
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3. a) Which angles inside or outside the star are
greater than 90 degrees?
b) Estimate the measure of these angles.
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4. For which sports or hobbies do you need to understand
rstand angles?
5. For which jobs or careers do you need to understand
and angles?
Career Link
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Jon is a carpenter who builds roof
trusses. A truss supports the weight
of the roof and determines the slope
of the roof. Jon needs to measure
angles properly to make accurate cuts.
Measuring accurately ensures that the
pieces will fit together well and the
structure will be strong.
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Literacy Strategy
Use a mind map to keep track of what
you know about angles.
Obtuse
Right
Acute
Straight
Types of
Angles
Reflex
Geometric Relationships • MHR 355
8 Get Ready
Number Skills
a)
b)
Determine each unknown value in #1 to
#3. Do as many as you can without using
a calculator.
1. a) 35 + 55 = b) 41 + 49 = c) 38 + = 90
d) 30 + 60 + 90 = c)
d)
e)
f)
e) 45 + + 90 = 180
2. a) 90 - 50 = b) 90 - 25 = c) 90 - = 15
180 - 110
d) _ = 2
e) 180 - = 75
180 - 30
f) _ = 2
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Measure Angles
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5. a) What is the measure of the angle
shown on the protractor?
3. a) 180 × 2 = b) 360 - 180 = c) 360 + 180 = d) 360 × 2 = H
b) Caleb incorrectly states that the
e) 360 × 3 = angle is 20°. Describe the error
Caleb made.
Estimate Angles
6. Write the angle measure shown on
4. Select the description that matches
each angle in parts a) to f).
A a bit less than 45°
B a bit less than 90°
C a bit more than 90°
D a bit less than 180°
E a bit more than 180°
F a bit less than 360°
356 MHR • Chapter 8
each protractor.
a)
E
b)
b)
F
c)
c)
G
d)
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H
e)
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d)
Construct Angles
9. Construct an angle that has each
measure.
a) 50°
c) 20°
b) 55°
d) 135°
Fractions
10. Draw a circle for each part and shade
the required fraction.
K
3
Example: _
4
7. Which, if any, of the angles in #6 are of
the following types?
a) acute angle
b) obtuse angle
c) right angle
d) straight angle
8. Use a protractor to measure each angle.
a)
1
a) _
8
_
c) 4
4
2
b) _
3
3
_
d)
8
Get Ready • MHR 357
8.1 Classifying and Estimating Angles
Minds on…
Learning Goals
I am learning to
• name different
types of angles
• develop personal
references for
certain angles
• estimate the
measure of
an angle
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Angle measures are important in many sports. How do you
think angles are involved in the sports shown? Do you think that
understanding angles would help you improve at any of these sports?
Action!
Tools
• masking tape
• marker
• protractor
(optional)
358 MHR • Chapter 8
Investigate Angles
1. Stick two 50-cm strips of masking
tape on the floor to make a plus sign.
Label 0°.
0°
2. Stand at the intersection of the strips and face 0°. Rotate
Literacy Link
Clockwise means
rotating in the
direction that the
hands on a clock
rotate.
Counterclockwise
rotation is the
opposite of
clockwise.
counterclockwise (to the left) until you are facing 0° again.
You have turned in a complete circle. How many degrees have
you turned?
3. Copy the table below into your notebook. Stand at the intersection
of the strips and face 0°. Rotate counterclockwise each fraction of
a turn in the first column of the table. Complete the columns titled
Angle Turned and Sketch of My Movement. The first fraction is
done for you.
Fraction
of a Turn
1
a) _
2
1
b) _
4
3
c) _
4
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1
d) _
8
Literacy Link
A personal reference
is an item you can
use to estimate
a measurement.
For example, the
distance from the
floor to a doorknob
is about 1 m.
Angle
Turned
Sketch of My
Movement
Personal
Reference
from 0° to 180°
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4. Reﬂect Spread your arms out wide. You can use this position as a
personal reference for a 180° angle. Choose a personal reference
for the other angles in step 3. Sketch or describe them in the
column titled Personal Reference.
5. Extend Your Understanding Use your personal references to
estimate the size of each of the following angles.
a)
b)
8.1 Classifying and Estimating Angles • MHR 359
Example 1: Classify and Estimate Angles
180° and Less
In a game of pool, a player strikes the white ball
so that it hits the black ball. The black ball travels
and sinks into a pocket. For parts a) to d) below,
the white and black balls start from the positions
in the diagram, and the black ball sinks into the
pocket indicated.
• Sketch the approximate angle that the paths of the white ball and
black ball make.
• Name the type of angle and estimate the angle measure.
a) blue pocket
b) green pocket
c) orange pocket
d) purple pocket
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Solution
a) blue pocket
Literacy Link
A straight angle
is 180°.
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The angle is a straight angle. The angle measure is 180°.
b) green pocket
An obtuse angle
is between 90°
and 180°.
The angle is an obtuse angle. The angle is about halfway between
90° and 180°. The angle measure is about 135°.
360 MHR • Chapter 8
c) orange pocket
Literacy Link
A right angle is 90°.
An acute angle is
between 0° and 90°.
The angle seems to be a right angle. The angle measure is 90°.
d) purple pocket
Processes
Communicating
How can you
represent the
angle between
the paths of
the balls if the
black ball sinks
into the yellow
pocket?
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The angle is an acute angle. The angle is between 45° and 90° and
is very close to 45°. The angle measure is approximately 50°.
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Show You Know 1
1. Classify each angle as either acute or obtuse.
a)
b)
c)
2. Use your personal references from the Investigate to estimate the
size of each angle in #1.
3. Measure each angle in #1. How close were your estimates to your
measurements?
8.1 Classifying and Estimating Angles • MHR 361
Example 2: Sketch Angles 180° and Greater
Rachel spends hours practising her
skateboard tricks.
a) Sketch the start and end positions
of a “180,” a jump in which the
skateboard makes a 180° turn.
b) Make similar sketches for a 270, a
360, and a 540.
Solution
In the diagrams, T stands for the toe of the board and H is for the heel.
1 of a full
a) In a 180, the toe of the skateboard rotates 180°, which is _
2
rotation.
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T
end
H H
T
start
3
4
its starting point. An angle greater than 180° but less than 360° is
called a reflex angle.
b) A 270 is _ of a full rotation. The toe of the board rotates 270° from
reflex angle
• an angle between
180° and 360°
225°
H
H
330°
T end
362 MHR • Chapter 8
T
start
Notice that
180° + 90° = 270°
and that
360° - 90° = 270°.
A 360 is 1 complete rotation. The position of the board at the end of
the jump is the same as the start position.
H
T
1 rotations. The
540 = 360 + 180. In a 540, the skateboard makes 1_
2
end position of the skateboard is the same as for a 180° jump.
540°
T
180°
end
H H
T
360°
start
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Show You Know 2
1. Classify each angle as either a straight angle or a reflex angle.
a)
b)
c)
2. Estimate the measure of each angle in #1. Explain your thinking.
3. Measure each angle in #1. How close were your estimates to your
measurements?
8.1 Classifying and Estimating Angles • MHR 363
Consolidate and Debrief
Key Ideas
•
•
•
•
•
•
•
An acute angle is between 0° and 90°.
A right angle is 90°.
An obtuse angle is between 90° and 180°.
A straight angle is 180°.
A reflex angle is between 180° and 360°.
One full rotation is 360°.
An angle greater than 360° involves a rotation greater than one full circle.
Apply
1. Choose one of the photos below. What types of angles can you see
in the photo?
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2. EQAO Which of the following angles is approximately 315°?
A
B
C
D
364 MHR • Chapter 8
3. While attempting a 180 on his skateboard, Dylan came up
T
a bit short.
a) Estimate the angle of the board relative to its starting position.
b) Measure the angle.
c) How close was your estimate to your measurement?
H H
T
4. Most building codes state that the angle between the ground and
deck stairs must be 30° or less. Building inspectors need to identify
angles that might not meet the code. For each photo, does the
angle the stairs make with the ground appear to be acceptable?
Explain your reasoning.
a)
b)
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5. Rhys designs a bike ramp.
a) Estimate ∠A.
b) Estimate ∠B.
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A
B
c) Is it possible to build a ramp where
the side of the ramp makes an obtuse angle
with the ground? Explain why or why not.
6. Achievement Check For safety, a ladder needs to form a 75° angle
with the ground.
a) Estimate the angle of the ladder in the diagram.
b) Is the ladder in the diagram at a safe angle? Explain why or
why not.
c) Measure the length of the hypotenuse in the diagram.
d) Measure the length of the base of the triangle.
e) Complete this statement:
The hypotenuse is times as long as the base of the triangle.
f) The ladder is moved to make a greater angle at the ground.
How will the lengths of the hypotenuse and base compare?
Literacy Link
In a right triangle,
the hypotenuse is
the side opposite
the right angle. It is
the longest side in a
right triangle.
8.1 Classifying and Estimating Angles • MHR 365
7. Patrick Chan is a Canadian figure skater. Like most
male figure skaters, he often includes at least one
quad in his program. A quad is a jump with 4 full
rotations. Through how many degrees does Chan
rotate during a quad?
Canadian figure skater, Patrick Chan
Reﬂect
R1. Look around your classroom. What type of angle seems to be the
most common? Why do you think this is so?
R2. Becky and Sonia are in construction class. Becky
says a saw cut is at a 135° angle. Sonia says the cut
is at a 45° angle. Explain how it is possible that
both students are correct.
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R3. You are making a round pillow with a wheel design. You have pieces
of fabric in different colours. Each piece is cut in the shape of an
equilateral triangle.
a) What is the measure of each angle of an equilateral triangle?
b) You use 4 triangles. How many more equilateral triangles will you
need to complete the pillow? How do you know?
c) You want to make a similar pillow even more colourful by using
more pieces of fabric. Describe three different triangles that you
could use to cover the pillow in a wheel design. Compare your
triangles with a classmate’s.
d) How can you verify that each of your triangles will cover the
pillow exactly?
366 MHR • Chapter 8
8.2 Angle Relationships of Triangles
Minds on…
Learning Goals
I am learning to
• determine the
relationship
among the
interior angles
of a triangle
• determine the
relationship
between the
interior and
exterior angles
of a triangle
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• determine the
relationship
among the
exterior angles
of a triangle
interior angle
• an angle formed
inside a figure by two
sides meeting at a
vertex
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Triangles are often used in construction because they form a stronger
structure than a rectangle. The angle made at the peak of a roof allows
rain and snow to slide off the sides. The more snow an area gets, the
steeper the roof needs to be.
• How many triangles do you see in these pictures?
• How many types of angles can you identify?
• Which triangles appear to be the same size?
• How does the interior angle at the peak affect the slope or pitch
of the roof?
8.2 Angle Relationships of Triangles • MHR 367
Action!
Investigate Interior Angles of Triangles
Is there a relationship among the three interior angles of a triangle?
Tools
• geometry
software
Method 1: Use Geometry Software
1. Open The Geometer’s Sketchpad®.
a) Select the Straightedge Tool. Draw a triangle.
b) Select the Arrow Tool. Highlight one vertex, including part of
each leg. From the Measure menu, select Angle. Repeat for the
other vertices.
2. From the Number menu, select Calculate. Click on one of the angle
measures. Then, click + in the calculator box. Add the other angle
measures and click OK. The sum of the interior angle measures will
be displayed.
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3. Drag a vertex to a different location.
Vertex A is
dragged down
and to the right.
368 MHR • Chapter 8
4. Reﬂect
a) What happens to the individual angle measures as you drag
one vertex?
b) What happens to the sum of the angle measures as you drag
one vertex?
5. Use a different vertex and repeat steps 3 and 4. Compare your
findings with a partner’s.
6. Extend Your Understanding
a) Write an equation that relates the sum of the interior angles, S,
and the individual angles of the triangle, A, B, and C.
b) Rewrite the equation from part a) to isolate one of the interior
angles. Substitute the value for S.
c) Is this a linear relationship? Explain why or why not.
Tools
• ruler or
straight edge
Method 2: Use Paper and Pencil
1. a) Work in a small group. Draw a triangle of any shape or size
different from those of your partners.
b) Label the vertices A, B, and C inside the triangle.
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2. Tear each angle off the triangle.
A
B
C
3. Place the pieces together so that
the vertices touch. What type of
angle do the pieces form?
C
A
B
4. Reﬂect
a) Evaluate ∠A + ∠B + ∠C for your triangle.
b) What is ∠A + ∠B + ∠C for each of your partners’ triangles?
5. Extend Your Understanding
a) Write an equation that relates the sum of the interior angles, S,
and the individual angles of the triangle, A, B, and C.
b) Rewrite the equation from part a) to isolate one of the interior
angles. Substitute the value for S.
c) Is this a linear relationship? Explain why or why not.
8.2 Angle Relationships of Triangles • MHR 369
Example 1: Determine an Interior Angle Measure of
a Triangle
What is the measure of ∠L?
J
30°
K
80°
L
Solution
The three interior angles of any triangle add to 180°.
∠J + ∠K + ∠L = 180°
30° + 80° + ∠L = 180°
110° + ∠L = 180°
110° + ∠L - 110° = 180° - 110°
∠L = 70°
The measure of ∠L is 70°.
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Show You Know 1
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1. Determine the size of each unknown interior angle.
a)
b)
A
F
40°
25°
B
80°
C
75°
G
c) L
H
d) U
30°
31°
125°
V
110°
M
370 MHR • Chapter 8
K
X
Example 2: Determine an Exterior Angle of a Triangle
exterior angle
• an angle formed
outside a figure
between one side
of the figure and
the extension of an
adjacent side
In PQR, side QR is extended to point S to create an exterior angle.
Determine the size of ∠PRS.
∠PRS refers to the
angle from P to R to
S. This notation is
used when a vertex
has more than one
angle.
P
47°
65°
Q
R
S
Solution
Method 1: Use a Two-Step Solution
Step 1: Two angles inside a triangle are known. Determine the size of
the third angle.
∠PRQ + ∠Q + ∠P = 180°
∠PRQ + 65° + 47° = 180°
∠PRQ + 112° = 180°
∠PRQ + 112° - 112° = 180° - 112°
∠PRQ = 68°
Step 2: The interior and exterior angles at R form a straight angle.
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∠PRQ + ∠PRS = 180°
68° + ∠PRS = 180°
68° + ∠PRS - 68° = 180° - 68°
∠PRS = 112°
The measure of ∠PRS is 112°.
Processes
Connecting
How can the
torn angles you
placed in the
Investigate help
you visualize
what ∠PRS
equals?
The measure of a
straight angle is 180°.
Method 2: Use a One-Step Solution
∠P + ∠Q + ∠PRQ = 180°
and ∠PRS + ∠PRQ = 180° .
Therefore, ∠P + ∠Q = ∠PRS.
∠PRS = ∠P + ∠Q
∠PRS = 47° + 65°
∠PRS = 112°
The measure of ∠PRS is 112°.
8.2 Angle Relationships of Triangles • MHR 371
Show You Know 2
1. a) Determine the measure of ∠ACB.
A
80°
60°
B
C
D
b) Identify a straight angle in the diagram.
c) What is the measure of ∠ACD? How do you know?
2. a) What is the sum of the measures of ∠U and ∠V?
U
70°
V
W
X
b) Identify an angle that has the same size as your answer to part a).
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3. Use the method of your choice to determine the size of each
unknown angle.
a) ∠ACD
A
40°
B
80°
C
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b) ∠FGE
E
D
c) ∠NKM
F
25°
75°
G
H
d) ∠WXV
L
U
30°
31°
125°
V
110°
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K
N
372 MHR • Chapter 8
X
W
Example 3: Properties of Exterior Angles of Triangles
Is the sum of the exterior angles of a triangle always the same?
Solution
Sketch a diagram of a triangle and include exterior angles. Label
the interior and exterior angles with variables.
q
c s
a
r
b
For each exterior angle, substitute
(180° - interior angle). Then, simplify.
Sum of exterior angles = a + b + c
= (180° - q) + (180° - r) + (180° - s)
= 540° - q - r - s
Factor -1 out of the
last three terms.
= 540° - 1(q + r + s)
= 540° - 1(180°)
What is the sum of
q + r + s? How do
= 540° - 180°
you know?
= 360°
Processes
Reflecting
How else could
you solve this
problem?
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Yes, the sum of the exterior angles of a triangle is always 360°.
This value is a constant and is
not specific for any triangle.
Show You Know 3
1. Two of the exterior angles of a triangle are 120° and 135°.
a) What is the size of the other exterior angle? How do you know?
b) What are the sizes of the interior angles?
2. Identify two sets of angles in the diagram that add to each measure.
x
t
y u
v
z
a) 180°
b) 360°
8.2 Angle Relationships of Triangles • MHR
373
Consolidate and Debrief
Key Ideas
•
B
•
∠A + ∠B + ∠C = 180°
To form an exterior angle, extend one side of a triangle.
The size of an exterior angle is equal to the sum of the
two interior angles opposite it.
Y
∠WYZ = ∠W + ∠X
The three exterior angles of any triangle add to 360°.
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c
b
a + b + c = 360°
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Apply
1. a) Without measuring, classify the angle at the peak
of the garage roof as acute, obtuse, or right.
b) Estimate the measure of the angle at the peak.
c) The front of the garage roof looks like an isosceles
triangle. If the angles at the base of the roof are each
35°, determine the angle at the peak.
2. Determine the measure of g in each diagram.
a)
b)
g
70°
144°
g
130°
3. EQAO What is the value of x in TUV?
T
x
A 70°
B 110°
C 250°
D 290°
374 MHR • Chapter 8
U
40°
C
W
X
•
A
The three interior angles of any triangle have a sum of 180°.
30°
V
Z
4. EQAO What is the size of ∠Z in the triangle shown?
X
110°
A 110°
B 70°
Y
C 40°
Z
D 35°
5. Deljaye makes a pair of earrings in the shape of an isosceles triangle.
a) She wants the base of the triangle to hang horizontally. Where
could she attach the ear post? Explain your reasoning.
b) What is the measure of each interior angle? How do you know?
c) Deljaye makes a new pair of earrings in the shape below. What is
the name for the type of triangle shown? Write an equation that
relates the sum of the interior angle measures, S, to the measure
of each interior angle, x.
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6. Achievement Check
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a) With or without technology, create a triangle. Extend one side
at each vertex to make three exterior angles. Label the interior
angles x, y, and z.
b) Express each exterior angle in terms of the interior angles.
c) What is the sum of the exterior angles? How do you know?
P
7. Alice is asked to measure the interior angles of the triangle
shown and to find their sum. Her solution is below.
∠P + ∠Q + ∠R = 47° + 93° + 46°
= 186°
This triangle has interior angles that add up to 186°.
a) Is there an error in Alice’s work? How do you know?
Q
R
b) If so, explain where you think the error occurred.
8. a) Sketch a triangle with angles 60°, 70°, and (2x + 10)°.
b) What is the value of x?
c) Show that your answer is correct.
8.2 Angle Relationships of Triangles • MHR 375
V
9. Solve for a.
2a
X
a
3a
W
10. Look at the triangle at the bottom
Processes
of the fence. What is the size of
each interior angle? Justify your
answer.
Reﬂect
Reasoning and
Proving
Use what you
know about
angles and
intersecting
lines.
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R1. Can a triangle have two right angles? Explain why or why not.
R2. a) In the diagram, which angle needs to be determined first?
Explain why.
b) State two ways to solve for the other unknown angles.
c) Use one of your methods from part b) to solve for the
unknown angles.
R3. Jaden and Steve are asked to copy a triangle and draw the
exterior angles. They make the following sketches.
Steve’s sketch
Jaden’s sketch
Jaden says Steve’s sketch is incorrect and that the exterior angles
must create a spiral. Did Steve make an error in his drawing? Explain
your reasoning.
376 MHR • Chapter 8
a
b c
75°
8 Mid-Chapter Quiz
Choose the best answer.
1. What type of angle is a 20° angle?
A acute
B obtuse
C reflex
D right
2. EQAO What is the better estimate for the interior
angles of the triangle shown?
A 20°, 60°, 100°
B 20°, 70°, 90°
C 30°, 50°, 90°
D 30°, 70°, 90°
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3. Each side of a triangle is extended at one of the vertices to make
three exterior angles. What is the sum of the exterior angles?
A 180°
B 270°
C 360°
D 720°
4. EQAO What is the measure of ∠E?
D
A 60°
80°
B 50°
C 40°
D 30°
140°
F
E
G
5. EQAO What is the value of x?
30°
A 18°
B 30°
C 48°
D 51°
48°
x
Mid-Chapter Quiz • MHR 377
8.3 Angles of Quadrilaterals
and Other Polygons
Minds on…
Learning Goals
I am learning to
• determine the
sum of the
interior angles
in all types of
polygons
• determine
interior angle
measures of
polygons
• determine
exterior angle
measures of
quadrilaterals and
other polygons
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Everything around us involves shapes.
• What shapes do you see in the tile pattern?
• How would you describe the shape of the table?
• How would you classify the shapes in the
other photos?
Action!
polygon
• a 2-D closed figure
made of three or
more line segments
378 MHR • Chapter 8
Investigate Interior Angles of Polygons
Is there a relationship between the sum of the interior angles of a
polygon and the number of sides?
Tools
• dynamic
geometry
software or
ruler and
protractor
1. a) With or without technology, create
quadrilateral ABCD.
b) Draw one diagonal that cuts the
quadrilateral into triangles.
c) How many triangles did you create?
d) What is the sum of the interior angles
of the quadrilateral? Justify your
answer.
B
C
A
D
2. Copy the table. Use the information from step 1 and fill in the values
for a quadrilateral.
Number
of Sides
Number of
Triangles Created
Sum of Interior
Angles
Triangle
3
1
180°
Quadrilateral
4
Polygon
Pentagon
Hexagon
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Decagon
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Sam
3. Repeat step 1 for a pentagon (5-sided figure) and a hexagon (6-sided
figure). Draw diagonals from one vertex to divide each shape into
the least number of triangles. Record your answers in the table.
4. Reﬂect
a) What is the relationship between the number of sides of a
polygon and the number of triangles within it?
b) Without sketching, how many triangles can be made inside an
octagon (8-sided figure)?
c) What would be the sum of the interior angles of an octagon?
d) Make a sketch to verify your answers to parts b) and c).
5. Extend Your Understanding
a) Describe the relationship between the sum of the interior angles
of a polygon, S, and the number of sides, n, of the polygon. Write
a formula to represent this relationship.
b) Determine the sum of the interior angles of a decagon (10-sided
figure).
8.3 Angles of Quadrilaterals and Other Polygons • MHR 379
Example 1: Determine Interior Angles of Polygons
For each sign determine the sum of the interior angles and the size of
each unknown interior angle.
a)
b)
STOP
Solution
regular polygon
• a closed figure in
which all of the
angles are equal, and
all of the sides are the
same length
Processes
Reasoning and
Proving
How do you
know the angles
at the outside
edge of each
triangle are
equal?
a) A stop sign is a type of regular polygon called a regular octagon.
Method 1: Use Properties of Regular Polygons
Draw a line from the centre of a regular octagon to each vertex.
This creates 8 identical isosceles triangles.
1 of a full rotation.
Each angle at the centre of the octagon is _
8
1 × 360° = 45° How many degrees
_
8
is a full rotation?
e
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Sam
One angle in each triangle is 45°. The sum
of the other 2 angles is 180° - 45°, or
135°. The angles at the outside edge of
135° or 67.5°.
each triangle are each _
2
Two adjacent 67.5° angles make up
one interior angle of the octagon. So,
the measure of each of the 8 interior
angles is 135°.
The sum of the interior angles is 8 × 135°,
or 1080°.
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67.5°
67.5°
67.5°
45°
Method 2: Use a Formula
To determine the sum of the interior angles of a polygon, S, use
the formula, S = 180°(n - 2), where n is the number of sides.
S = 180°(n - 2)
S = 180°(8 - 2) An octagon has 8 sides.
S = 180°(6)
S = 1080°
The sum of the interior angles in an octagon is 1080°.
In a regular octagon, all 8 interior angles are equal.
1080° = 135°
_
8
Each interior angle of a stop sign is 135°.
380 MHR • Chapter 8
67.5°
b) The school crossing sign is a pentagon, but not a regular pentagon.
So, use the formula.
S = 180°(n - 2)
S = 180°(5 - 2)
S = 180°(3)
S = 540°
The sum of the interior angles in a pentagon is 540°.
Three of the angles are labelled as right angles. The other two angles
are identical.
Let a represent the size of each unknown angle.
Sum of the angles = 3(90°) + 2a
540° = 270° + 2a
540° - 270° = 270° + 2a - 270°
270° = 2a
2a
270° = _
_
2
2
135° = a
The size of each unknown interior angle is 135°.
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Show You Know 1
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1. Determine the sum of the interior angles of each polygon. Then, determine
the size of each interior angle.
a) regular hexagon
b) square
c) regular pentagon
2. Without calculating, estimate the size of each unknown angle.
a)
b)
122°
b
a
c)
140°
110°
c
100°
95°
d)
35°
d
80°
110°
125°
130°
d
3. Determine the measure of the unknown angles in #2. Compare your
answers with your estimates.
8.3 Angles of Quadrilaterals and Other Polygons • MHR 381
Example 2: Exterior Angles of Polygons
Determine the sum of the exterior angles for each polygon.
a)
b)
110°
73°
87°
50°
Solution
a) Identify the unknown exterior angles and determine their size.
180° - 73° = 107°
and
180° - 87° = 93° What is the sum of the
angles in a straight angle?
Sum of exterior angles = 110° + 93° + 50° + 107°
= 360°
The sum of the exterior angles of the quadrilateral is 360°.
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b) Method 1: Use a Protractor
Copy the diagram of the regular
hexagon and draw one exterior
angle.
Measure the exterior angle and
multiply by 6.
60°(6) = 360° Why do you
not need to
measure all six
exterior angles?
The sum of the exterior
angles is 360°.
382 MHR • Chapter 8
Method 2: Use the Interior Angle Measure
Use the formula, S = 180°(n - 2), where S is the sum of the interior
angles, and n is the number of sides.
S = 180°(6 - 2)
S = 180°(4)
S = 720°
Divide the sum by 6 to find the size of each interior angle.
720° = 120°.
Each interior angle is _
6
Exterior angle = straight angle - interior angle
= 180° - 120°
= 60°
The sum of the exterior angles is 6 × 60°, or 360°.
Show You Know 2
1. Determine the sum of the exterior angles for a regular pentagon.
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2. a) What is the sum of the exterior angles for a 12-sided polygon?
How do you know?
b) Determine the size of each exterior angle of a 12-sided regular
polygon.
Consolidate and Debrief
Key Ideas
•
•
•
The sum of the interior angles of a polygon depends on the number of sides.
You can determine the sum of the interior angle measures, S, using the formula
S = 180°(n - 2), where n is the number of sides of the polygon.
The sum of the exterior angles of any polygon is 360°.
8.3 Angles of Quadrilaterals and Other Polygons • MHR 383
Apply
1. a) A hex head nut is in the shape of a regular hexagon. The picture
1 -inch hex head nut. What is the measure of each
shows a _
2
interior angle?
3
b) Determine the measure of each interior angle for a _-inch hex
4
head nut.
c) What do you notice about the answers to parts a) and b)?
Explain why this is so.
2. Vyaz is an estimator for a glass company. He measures
the glass required and tells customers the price.
a) Is the window shown a regular pentagon?
Explain why or why not.
b) The three interior angles at the top are all equal.
What is their size?
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3. EQAO Three interior angles of a quadrilateral are known.
a) Determine the measure of a. Explain your reasoning.
b) Determine b + c + d + e.
c a
85°
d
b
105°
75° e
4. The inside edges of a loonie make a regular 11-sided figure.
a) Determine the sum of the interior angles of a loonie.
b) What is the measure of each interior angle?
5. Achievement Check A sketch of a backyard gazebo shows a
floor that is a regular pentagon. All floorboards need to be cut at
the same angle so the pieces fit together. At what angle does
each floorboard need to be cut?
384 MHR • Chapter 8
6. EQAO What is the measure of each unknown angle?
117°
A x = 122°, y = 66°
y
122°
B x = 119°, y = 58°
C x = 117°, y = 48°
x
66°
D x = 114°, y = 58°
7. EQAO Determine the value of c.
78°
Justify your response.
2c
c
8. Estimate the size of the unknown interior angle of the arrowhead.
Then, calculate the angle measure.
40°
25°
40°
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9. EQAO A quilt has a star pattern of pieces in the
shape of a parallelogram. The pieces have an angle
of 45° at the centre. Stephanie’s grandmother
makes quilts with star patterns that have an angle
of 40° at the centre. How many points will her
stars have? Justify your answer.
45°
40°
Reﬂect
R1. Explain how to determine the sum of the interior angles
of a regular polygon, and the size of each angle.
R2. Describe two ways of finding the size of each exterior angle
of a regular polygon. Which method do you prefer? Why?
R3. a) What is similar about interior angles and exterior angles?
b) What is different about interior angles and exterior angles?
8.3 Angles of Quadrilaterals and Other Polygons • MHR 385
8.4 Angles of Parallel Lines
Minds on…
Learning Goals
I am learning to
• identify parallel
abeth
Way
ve.
dA
llan
We
Ni
ag
ar
aS
to
ne
Ont
ario
St.
McNab Rd.
n Eliz
St.
lton
r
a
C
Bunting Rd.
386 MHR • Chapter 8
Quee
t.
tt S
Sco
Niagara St.
transversal
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Sam
d.
Vine St.
transversal
• a line that crosses
two or more
parallel lines
Lake St.
parallel lines
• lines that are the
same distance apart
• lines that never
intersect
• often marked
with matching
arrowheads
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Geneva St.
involving parallel
lines
ll R
we
Lin
Grantham Ave.
• solve problems
.
ll Rd
Stewart Rd.
ne
Par
Rd.
Read Rd.
relationships
involving parallel
lines and a
transversal
ore
Rd
.
sh
Lake
Irvine Rd.
• recognize angle
Townline Rd.
lines and
transversals
St. Catharines
406
In part of St. Catharines, Ontario, the north-to-south streets do run
north and south, but the cross streets do not run east and west. They
are parallel to the shore of Lake Ontario. The result is that none of the
major intersections meet at right angles.
• Who might consider the angles at these intersections to be a
problem? Why?
• Would you have designed the streets like this? If not, how would you
have done it?
• Which streets can be represented by parallel lines?
• What street could be a transversal?
Action!
Investigate Angle Relationships With Parallel Lines
Tools
• tracing paper
• protractor
1. a) Fold a piece of paper on a diagonal near
opposite angles
• angles formed on
either side of two
lines that intersect
the middle of the paper. Unfold the paper.
b) Fold the top of the paper straight down
to create a horizontal crease. Unfold
it. Fold the paper straight down again
U
to create another horizontal crease
about 5 cm from the other crease.
W
c) Write labels on your paper that match
the diagram. Line segment YZ is a
transversal, crossing both UV and
WX. When a transversal crosses two
parallel lines, eight angles are created.
Y
k h
m n
V
q r
s t X
Z
2. a) Predict which of the eight angles are
x
y
x=y
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corresponding
angles
• angles that are
in the same
position relative to
parallel lines and a
transversal
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Sam
the same size.
b) Use tracing paper to check your answer.
3. a) Use a protractor and measure k.
b) Without measuring, determine the size of h.
c) Use your answers to parts a) and b) to determine
the size of m and n.
d) Identify two pairs of opposite angles.
4. Reﬂect
x
a) What do you think is the size of q? How do you know?
z
b) What do you think are the sizes of r, s, and t?
x=z
c) Identify four pairs of corresponding angles.
co-interior angles
• angles formed
between parallel
lines, on the same
side of the transversal
x
Corresponding
angles form an F.
5. Extend Your Understanding
a) Angles n and r are called co-interior angles.
What is the sum of n and r?
b) Identify another pair of co-interior angles.
Co-interior
angles form a C.
y
x + y = 180°
8.4 Angles of Parallel Lines • MHR
387
Example: Angle Measures Formed by a Transversal
Spencer drives a large food delivery
First
Street
truck. Not all of the streets meet at
90° angles. So, Spencer plans his
route to make sure his truck can
make the turns.
a b
Reading his map, Spencer sees that
c d
First Street and Third Street look
parallel. He sees that Fourth Avenue
looks like a transversal. He estimates
that the right turn from Fourth
Avenue to First Street is about 80°.
Estimate the size of all possible turns at the intersections
of Fourth Avenue with First Street and Third Street.
alternate angles
• angles formed
between parallel
lines, on opposite
sides of the
transversal
w
y
w=y
Third
Street
e f
g h
Solution
c is about 80°.
b and c are opposite angles. So, b is also about 80°.
f corresponds to b, and g corresponds to c. So, f and g are each about 80°.
a is about 100° because a + c = 180°. So, d, h, and e are also each
about 100°.
Angles d and e are called alternate angles. They are equal.
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Alternate angles form a Z. Which
other angles are alternate angles?
Show You Know
1. Without measuring, identify all of the angles that are 70°.
a c
b d
e g
f 70°
2. Without measuring, determine the size of each unknown
angle. Explain your reasoning.
g 112°
h i
388 MHR • Chapter 8
Fourth
Avenue
j m
k n
Consolidate and Debrief
Key Ideas
•
•
•
Parallel lines are always the same distance apart. Therefore, they never intersect.
A line that crosses two or more parallel lines is called a transversal.
For the angles formed by a transversal and parallel lines,
opposite angles are equal.
X
X
F
corresponding angles are equal. (They form an F.)
F
Apply
co-interior angles add to 180°. (They form a C.)
C
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Sam
alternate angles are equal. (They form a Z.)
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C
Z
Z
1. These lines are clearly not parallel, yet they
do not intersect. Explain why.
2. Determine the measure of each angle indicated.
a)
b)
60°
c
c)
100°
f
d)
z
48°
79°
x
8.4 Angles of Parallel Lines • MHR 389
Marsdale Drive
3. Glendale Avenue and Marsdale Drive are perpendicular.
What is the size of each angle at the intersection?
Glendale Ave.
4. The marine signal flag for a person overboard is a rectangular flag.
Which other angle(s) are 60°?
a
c
b
e
d
60°
5. A town plans its bus routes based on the angles the streets make. For
Processes
a turn to be on a bus route, it must be 80° or greater. Victoria Avenue
and Whitehorse Road are parallel.
Victoria
Avenue
Halifax
Drive
Whitehorse
Road
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95° a
e f
bc
g h
j
Connecting
e
l
p
Sam
d
k
What other
decisions in a
town might
be affected by
intersection
angles of
streets?
30°
n
m
q
p
Winnipeg
Street
55°
r
s t
a) Identify the turns that could be included on a bus route.
b) Open Question Plan a bus route between any of the locations
identified with stars.
6. EQAO Which statement is not true for the diagram?
A w=z
B x=y
x
w
C x = 90° - z
y
D y + z = 180°
z
390 MHR • Chapter 8
7. Achievement Check Jessie creates a mosaic from pieces of
q r
coloured glass. She determines the angles at which each
piece of glass will be cut.
a) What is the sum of any three adjacent angles at the
centre? Explain why.
b) What is the sum of all six angles at the centre?
Explain why.
c) Suppose a = 65° and b = 50°. Determine the measure
of the unknown angles. Justify your answers.
s t
b
a c
d e f
u v
w x
8. EQAO What is the measure of x in the diagram?
A 133°
130°
B 130°
x
C 90°
47°
D 83°
9. Determine the measure of each unknown angle.
e
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Sam
40°
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w x
y z
a c
b
e f
g h
q
t r
s
Reﬂect
R1. Jodi studies her answers to #9. She notices letter patterns in
the angles.
a) Describe what is true about angles that form a C.
b) Describe what is true about angles that form an F.
c) Describe what is true about angles that form a Z.
R2. How can you test whether these line segments are parallel?
R3. In the diagram, a = 118° and g = 61°. Are line segments PQ
P
R
and RS parallel? Explain your reasoning.
a b
e f
Q
c d
g h
S
8.4 Angles of Parallel Lines • MHR 391
8 Review
Learning Goals
Section
After this section, I can
8.1
8.2
name different types of angles
develop personal references for certain angles
estimate the measure of an angle
determine the relationship among the interior angles of a triangle
determine the relationship between the interior and exterior angles
of a triangle
determine the relationship among the exterior angles of a triangle
8.3
8.4
determine the sum of the interior angles in all types of polygons
determine interior angle measures of polygons
determine exterior angle measures of quadrilaterals and other polygons
e
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Sam
identify parallel lines and transversals
recognize angle relationships involving parallel lines and a transversal
solve problems involving parallel lines
9
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If you are unsure about any of these questions, review the appropriate section
or sections of this chapter.
8.1 Classifying and Estimating Angles, pages 358–366
1. Classify each angle.
a)
b)
c)
d)
2. a) Estimate the measure of each angle in #1.
b) Measure each angle in #1. How close were your estimates?
392 MHR • Chapter 8
8.2 Angle Relationships of Triangles, pages 367–377
3. Determine the measure of each unknown angle.
a)
b)
a
60°
c)
100°
c d
d)
2x
35°
r
65°
b 95°
5x
s
t
3x
4. a) Determine the size of b.
b
b) What is the sum of the exterior angles of any triangle?
160°
115°
8.3 Angles of Quadrilaterals and Other Polygons, pages 378–385
5. a) What is m + n? How do you know?
b) What is j + k + m + p?
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92°
k
95° j
e
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p
Sam
c) What is the sum of the four exterior angles?
p
d) Determine the measure of each unknown angle.
46°
m
n
6. EQAO Elias builds a rectangular platform for the porch of a
house. The diagonal support beams create four isosceles triangles.
Determine the values of x and y. Justify your answers.
x
y
8.4 Angles of Parallel Lines, pages 386–391
37°
7. The diagram shows parallel lines and a transversal.
a b
c d
a) List all the acute angles.
b) List all the obtuse angles.
e f
g h
c) a = 70°. Determine the measure of each of the other seven angles.
Justify your answers.
8. Are KL and MN parallel? How do you know?
K
L
48°
35°
M
100°
N
Review • MHR 393
8 Test Yourself
Multiple Choice
For #1 to #5, select the best answer.
1. Which type of angle is not shown on the diagram?
Hint: Look inside and outside the triangle.
A acute
B obtuse
C reflex
D right
2. For PQ and RS to be parallel, what must be the measure of x?
A 75°
B 85°
C 95°
D 105°
3. What is y + z?
A 90°
B 180°
C 270°
P
95°
R
x
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y
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Q
S
z
D 360°
4. How many other angles in the diagram are the same size as a?
A 0
B 1
C 2
D 3
5. Which expression represents the sum of the interior angles of a
regular hexagon?
A 180° × 4
B 180° × 6
C 180°(2 - n)
D 180°(n + 2)
394 MHR • Chapter 8
d a
c b
h e
g f
Open Response
6. The diagram shows the marine signal that means, “Require
d
assistance.” The flag is in the shape of a rectangle. Determine
the measure of all unknown angles.
a
b
e
f
j
g
h
m
k
7. A heptagon is a 7-sided figure.
110°
i
a) Calculate the sum of the interior angles for a heptagon.
b) Determine the size of each interior angle for a regular heptagon.
8. Randi works as a fitness trainer. She is setting up the ring used
for mixed martial arts. What is the measure of each interior
angle of a regular octagon?
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9. a) Solve for x.
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p
Sam
r
b) Determine the measure of each interior angle.
x
c) What is the measure of the exterior angles r, s, and t?
s 6x
2x
t
10. Determine the measure of each unknown angle.
p
85°
q
s
t
r
68°
u
11. EQAO Determine the values of x and y.
x
Justify your answers.
y
46°
57°
Test Yourself • MHR 395
8 Chapter Project
Logo Design
The universal recycling symbol looks like an equilateral
triangle. It was designed in 1970 by Gary Anderson.
He may have used a triangular shape because of the
three parts of environmental awareness: reduce, reuse,
and recycle.
• Design a logo that has a symbolic shape and uses
three of the topics of this chapter. For example,
consider triangles, quadrilaterals, other polygons,
parallel lines, and transversals. Create your logo
using one of the following:
– computer software
– pencil and paper
– craft materials
• Describe how the shape of your logo connects to
what it represents.
• Calculate all of the interior and exterior angles in
your design using skills learned from this chapter.
Use a protractor to measure any angles that cannot
be determined using another method.
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396 MHR • Chapter 8
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Sam
Games and Puzzles
Tiling Game
8
Tools
• one piece of
The object of the game is to place tiles on a playing board so that they
fit together at appropriate angles
Bristol board
per pair of
students
• plastic
shape tiles
(40 hexagons,
50 squares,
60 triangles)
• ruler
• glue (optional)
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1. Play the game with a partner. On the Bristol board, draw a playing
space that is 25 cm by 25 cm.
2. Decide who plays first.
a) The first player places a tile anywhere in the square
playing space.
b) The next player places a tile such that it touches at least one
side of tiles already played. Make sure the tile does not overlap
any tiles or go outside the square playing space.
c) As each tile is placed, the player records a number of points
equal to the number of sides on the tile.
3. Play continues as each player places one tile at a time.
4. If a player places a tile and creates an empty space, the player
loses 3 points. Players cannot skip a turn to avoid creating an
untiled gap.
5. The game ends when no more tiles can be placed. The player
with the highest score wins.
The square tile
creates an empty
space, so the player
loses 3 points.
Games and Puzzles • MHR 397