Download Triangles

7.1
Try These e.g.,
Triangles
Try These
You will need
• string
• scissors
• plain paper
• a millimetre ruler
• a protractor
1. e.g.,
Make a paper triangle. Draw a dot at each vertex. Cut the triangle so that
each vertex is separate. Show that the sum of the angles is 1808.
Cut three pieces of string that you can use to make a triangle.
How many different triangles can you make?
1
Place your string on paper to make a triangle. Mark the
vertices with a pencil. Join the vertices.
2
175 mm, and 184 mm
What are the side lengths? e.g., 128 mm,C07-F14-AW12.ai
3
What are the angle measures? e.g., 668, 748, and 408
4
Two triangles are different if they are not congruent. Are any
no
different triangles possible with your side lengths?
5
Compare your triangle with other students’ triangles. Could
anyone make more than one triangle? no
Reflecting
Suppose that
the sum of the
lengths of the
two shortest
sides is less than
the length of the
longest side. Can
these three pieces
of string make a
triangle? Explain.
C07-F114-AW12.ai
property
a characteristic
that is shared by
all the members
of a group
Example 1
The bamboo stems in this photograph create
an isosceles
triangle. An isosceles triangle
AW12
has 0176519637
two equal sides called legs. The interior
Figure
Number the C07-F14-AW12.ai
angles
opposite
legs are also equal.
Company
B
MPS
Do all
isosceles triangles have these
Technical
properties?
1st pass
Pass
Approved
Solution
Not Approved
A. Find the midpoint of side AC.
Label it M. Draw MB.
A
M
C
B. What are the side lengths, in millimetres?
nABM: 19 mm, 50 mm, and 47 mm
nCBM: 19 mm, 50 mm, and 47 mm
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C. Is nABM congruent to nCBM? How do you know?
Yes. e.g., They are congruent because only one triangle is
possible with these sides. OR They are the same size and
shape.
D. e.g.,
D. Kate said that this property is a property of all isosceles
E
triangles. Do you agree with Kate? Explain. Include a diagram.
• The angles opposite the equal legs are equal.
e.g., Yes, I agree. If you draw a centre line, you get two
congruent triangles. So the corresponding angles are equal.
20 mm
D
20 mm
45°
M 45°
35 mm
F
Example 2
Pavlo is a carpenter. He uses triangular brackets for shelving.
The sides of each bracket extend past the vertices to create
exterior angles. What types of triangles have this property?
• Each exterior angle is 908 or greater.
Solution
A. Measure the interior and exterior angles in the acute triangle
below. Record the angle measures on the diagram.
Acute triangle:
90°
C07-F15a-AW12.ai150°
120°
Obtuse triangle:
e.g.,
105°
152°
75°
35°
70°
110°
145°
28°
132°
48°
20°
160°
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B. Draw an obtuse triangle in0176519637
Part A. Extend one side at each
Figure Number
C07-F15a-AW12.ai
vertex to create three exterior angles. Measure the interior
Company
MPS
and exterior angles. Record
the measures. Are any exterior
Technical
angles acute? yes
Pass
1st pass
Approved
C. Is the following a propertyNot
ofApproved
all triangles? Explain.
• Each exterior angle is 908 or greater.
No. e.g., One exterior angle
than 908.
C07-F18-AW12.ai
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on0176519637
the obtuse
FN
CO
Technical
D. What triangles have the property in Part C?
acute triangles and right
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07_AW12_Ch07.indd 165
Why is showing
that something
is not a property
easier than
showing that it is
a property?
triangle is less
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CrowleArt Group
Pass
Approved
triangles
Not Approved
Reflecting
2nd pass
Hint
Use the triangular
bracket above as
an example of a
right triangle.
Chapter 7 Polygons
165
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Practice
Reflecting
Does it matter
which side of
a triangle you
extend to make
an exterior angle?
Explain.
1. Use your triangles from Example 2.
a) The sum of the interior angle plus the exterior angle is the
same at each vertex. What is this sum? 1808
b) Why does it make sense that each vertex has the same sum?
When you extend one side, you create two angles that
form a straight line . Angles that form a straight line
have a sum of 1808 .
c) Is this a property for all triangles? Explain.
• The sum of the interior angle plus the exterior angle
is 1808.
Yes. e.g., You always create an exterior angle by extending
a side. The interior angle and exterior angle will always
form a straight line.
Hint
Use the diagrams
and definitions of
different types of
triangles in Getting
Started.
2. Cables on the Esplanade Riel Bridge in Winnipeg illustrate
many types of triangles.
Circle the types of triangles that have each property.
a) Some sides are equal.
equilateral triangle
isosceles triangle
scalene triangle
b) Some exterior angles are equal.
equilateral triangle
isosceles triangle
scalene triangle
c) No interior angles are equal.
equilateral triangle
isosceles triangle
scalene triangle
d) All three exterior angles are 908 or greater.
acute triangle
obtuse triangle
right triangle
e) Each exterior angle is equal to the sum of the interior
angles at the other two vertices.
acute triangle
obtuse triangle
right triangle
3. a) What is one property of isosceles triangles that is not a
property of all triangles?
e.g., Isosceles triangles have exactly two equal sides.
b) What is one property of isosceles triangles that is a
property of all triangles?
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e.g., The sum of the interior angles is 1808.
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4. Use the angle measures to calculate the unknown angles
in each triangle. Include interior angles and exterior angles.
Record the measurements on the diagrams.
75°
105° 43°
1
60°
120°
35° 125°
2
30°
137°
32°
20°
30° 150°
145°
160°
3
55°
150°
148°
5. Use the triangles in Question 4. Complete this chart.
Triangle
Sum of 3 interior
angles
Sum of 3 exterior
angles
1
1808
1808
1808
3608
3608
3608
2
3
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5408
5408
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5408
Marcel wonders about this question.
• Does drawing a line parallel to the base of any
triangle create a second triangle with angles that
are equal to those in the original triangle?
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Figure Number
C07-F20-AW12.ai
Company
MPS
Technical
1st pass
Pass
60°
60°
Not Approved
60°
60°
AW12 idea.
a) Test Marcel’s
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• Draw a triangle. Draw a line through your triangle so
Figure Number
C07-F22-AW12.ai
that the
line is parallelMPS
to the base.
Company
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Technicalin the small triangle equal to the angles in
MPSthe angles
Are
the largePass
triangle? yes 1st pass
1st pass
Approved
Do you think that
the sum of the
interior angles and
the exterior angles
is the same for all
triangles? Explain.
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6. Marcel’s crew builds A-frame cabins in Tofino.
• The balcony is parallel to the base of a cabin.
• The front of this cabin is an equilateral triangle.
• The section above the balcony is also an
equilateral triangle.
AW12
Reflecting
Sum of 3 interior
angles 1 sum of
3 exterior angles
b) CompareApproved
your results with a classmate’s results.
Not Approved
Did your classmate get the same results? yes
c) Will adding a line that is parallel to the base always create
a smaller triangle with the same angles? Explain.
60°
Hint
One way to draw
parallel lines is to
draw along both
sides of a ruler.
6. a) e.g.,
65°
25°
25°
Yes. e.g., One angle is shared by both triangles. The other two angles are corresponding
angles, formed by transversals that meet the parallel lines at the same angle. So each
angle in the small triangle
has a matching equal angle in the large triangle.
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Chapter 7 Polygons
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FN
CO
Technical
167
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