7.1 Triangles Try These e.g., - Grade 12 Essentials Math · 2019-09-27 · 3 55° 20° 5. Use the...
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Triangles 7 . 1 You will need • string • scissors • plain paper • a millimetre ruler • a protractor Try These 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. 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. A C B M 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 What are the side lengths? 3 What are the angle measures? 4 Two triangles are different if they are not congruent. Are any different triangles possible with your side lengths? 5 Compare your triangle with other students’ triangles. Could anyone make more than one triangle? Example 1 The bamboo stems in this photograph create an isosceles triangle. An isosceles triangle has two equal sides called legs. The interior angles opposite the legs are also equal. Do all isosceles triangles have these properties? Solution A. Find the midpoint of side AC. Label it M. Draw MB. B. What are the side lengths, in millimetres? nABM: nCBM: property a characteristic that is shared by all the members of a group e.g., 128 mm, 175 mm, and 184 mm e.g., 668, 748, and 408 no Try These e.g., no 1. e.g., 19 mm, 50 mm, and 47 mm 19 mm, 50 mm, and 47 mm 164 Apprenticeship and Workplace 12 NEL
Transcript of 7.1 Triangles Try These e.g., - Grade 12 Essentials Math · 2019-09-27 · 3 55° 20° 5. Use the...
Triangles7.1Youwillneed• string• scissors• plain paper• a millimetre ruler• a protractor
Try TheseMake 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.
ReflecTinGSuppose 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.
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A C
B
M
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 What are the side lengths?
3 What are the angle measures?
4 Two triangles are different if they are not congruent. Are any different triangles possible with your side lengths?
5 Compare your triangle with other students’ triangles. Could anyone make more than one triangle?
Example 1The bamboo stems in this photograph create an isosceles triangle. An isosceles triangle has two equal sides called legs. The interior angles opposite the legs are also equal.
Do all isosceles triangles have these properties?
SolutionA. Find the midpoint of side AC.
Label it M. Draw MB.
B. What are the side lengths, in millimetres?
nABM:
nCBM:
property
a characteristic that is shared by all the members of a group
e.g., 128 mm, 175 mm, and 184 mm
e.g., 668, 748, and 408
no
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Try These e.g.,
no
1. e.g.,
19 mm, 50 mm, and 47 mm
19 mm, 50 mm, and 47 mm
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164 Apprenticeship and Workplace 12 NEL
07_AW12_Ch07.indd 164 02/03/12 12:25 PM
C. Is nABM congruent to nCBM? How do you know?
D. Kate said that this property is a property of all isosceles triangles. Do you agree with Kate? Explain. Include a diagram.• The angles opposite the equal legs are equal.
Example 2Pavlo is a carpenter. He uses triangular brackets for shelving. The sides of each bracket extend past the vertices to create exteriorangles. What types of triangles have this property?• Each exterior angle is 908 or greater.
SolutionA. Measure the interior and exterior angles in the acute triangle
below. Record the angle measures on the diagram.
Acute triangle: Obtuse triangle:
B. Draw an obtuse triangle in Part A. Extend one side at each vertex to create three exterior angles. Measure the interior and exterior angles. Record the measures. Are any exterior angles acute?
C. Is the following a property of all triangles? Explain.• Each exterior angle is 908 or greater.
D. What triangles have the property in Part C?
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150°
120°
90°
ReflecTinGWhy is showing that something
is not a property easier than
showing that it is a property?
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35° 110°
145°
70°
75°
105°
Use the triangular bracket above as an example of a right triangle.
Hint
Yes. e.g., They are congruent because only one triangle is
possible with these sides. OR They are the same size and
shape.
e.g., Yes, I agree. If you draw a centre line, you get two
congruent triangles. So the corresponding angles are equal.
yes
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D FM45° 45°
35 mm
20 mm 20 mm
E
D. e.g.,
e.g.,
No. e.g., One exterior angle on the obtuse triangle is less
than 908.
acute triangles and right triangles
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28°
160°
152°
132°20°48°
Chapter 7 Polygons 165NEL
07_AW12_Ch07.indd 165 02/03/12 12:25 PM
Practice 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?
b) Why does it make sense that each vertex has the same sum?
When you extend one side, you create angles that form . Angles that form have a sum of .
c) Is this a property for all triangles? Explain.• The sum of the interior angle plus the exterior angle
is 1808.
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?
b) What is one property of isosceles triangles that is a property of all triangles?
ReflecTinGDoes it matter which side of a triangle you
extend to make an exterior angle?
Explain.
Use the diagrams and definitions of different types of triangles in Getting Started.
Hint
1808
twoa straight line a straight line
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.
e.g., Isosceles triangles have exactly two equal sides.
e.g., The sum of the interior angles is 1808.166 Apprenticeship and Workplace 12 NEL
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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.
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137°
75°
148°
43°105°
1
32°
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60°
30°
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150°
2
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35°3
55°
20°
5. Use the triangles in Question 4. Complete this chart.
Triangle
Sumof3interior
angles
Sumof3exterior
angles
Sumof3interiorangles1sumof3 exteriorangles
1
2
3
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.
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?
a) Test Marcel’s idea. • Draw a triangle. Draw a line through your triangle so
that the line is parallel to the base.
Are the angles in the small triangle equal to the angles in the large triangle?
b) Compare your results with a classmate’s results. Did your classmate get the same results?
c) Will adding a line that is parallel to the base always create a smaller triangle with the same angles? Explain.
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60° 60°
60°60°
ReflecTinGDo you think that
the sum of the interior angles and the exterior angles is the same for all triangles? Explain.
One way to draw parallel lines is to draw along both sides of a ruler.
Hint
180818081808
360836083608
540854085408
yes
yes
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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6. a) e.g.,
Chapter 7 Polygons 167NEL
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