GR 9 3-5-5 LEARNERS
Page 1 of 4
NAME:
Gr 9
Date:
Time
1 hr.
3-5 Investigating properties of geometric figures: Investigate
CAPS
the angles in a triangle, focusing on the relationship between the
Reference
exterior angle of a triangle and its interior angles.
3-5-5 Investigate the angles in a triangle.
Topic
1.
Think First! [2 mins]
Tick any of these words that we usually associate with triangles.
equilateral square-angled rectangular scalene right-angled obtuse-angled
isosceles reflex-angled acute-angled equal-angled three-sided
2.
Go ahead! [13 mins]
2.1
Give TWO names for each of these kinds of triangles:
2.1.1
2.1.2
2.1.3
2.1.4
||
||
2.1.5
2.1.6
||
2.2
|
|
Calculate the sizes of the angles marked with a letter in these triangles.
e
a
c
29°
||
42°
d
63°
2.3
2.1.7
||
53°
What fact can you use to justify your solutions? Write a sentence starting:
I know that the ________________________________________________________________
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GR 9 3-5-5 LEARNERS
3.
Page 2 of 4
Check your work! [5 mins]
Check your answers to the naming of triangles and calculations.
4.
Think Again! [10 mins]
What was your answer to 2.3?
You may have written something like this: The sum of the angles of a triangle is 180°.
You can demonstrate this in a number of ways.
4.1
Draw 2 or 3 different looking triangles and measure their angles.
Add the measurements.
What do you notice?
4.2
- Draw a triangle on a piece of paper (advertising pamphlets are a good idea)
and neatly CUT it out.
- TEAR (not cut) off the angles.
- Paste the angles together with the CUT sides next to each other
(the torn sides are outside).
What do you notice?
Three angles pasted with
Three angles torn off
cut vertices together
The angles are adjacent angles on a straight line and
therefore are supplementary (add up to 180°)
5.
Think Some more! [15 mins]
5.1
What do the words interior and exterior mean?
When we work with the angles of a triangle,
we often refer to the interior and exterior angles.
The darker angles are inside the triangle.
We call them the interior angles
The lighter angles are outside the triangle.
We call them the exterior angles
5.2
5.3
Draw 2 or 3 different looking triangles and measure their interior and exterior angles.
See if there is a relationship between the exterior angles and the interior angles
opposite them.
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GR 9 3-5-5 LEARNERS
5.4
5.5
Page 3 of 4
Complete this sentence The exterior angle of a triangle is equal to the sum of
the ___ ___ ___ ___ ___ ___ ___ ___ ___ ___ ___
___ ___ ___ ___ ___ ___ ___ ___ angles.
Another way to demonstrate this fact:
- Draw a triangle on a piece of paper (advertising pamphlets are a good idea) and
neatly CUT it out.
- TEAR (not cut) off TWO of the angles.
- Paste the angles (cut vertices together) next to the third angle of the triangle.
What do you notice?
Two interior angles
torn off
Two interior angles pasted with cut vertices
together. The angles are opposite each other.
6.
Go ahead! [5 mins]
6.1
Calculate the sizes of the angles marked with a letter in these triangles.
158°
b
35°
d
c
a
||
47°
||
e
7.
Check your work! [5 mins]
Check your answers to the calculations.
Going further and even further!
Sierpinski Triangles –
Turn to the next page.
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GR 9 3-5-5 LEARNERS
Page 4 of 4
8.
Going further!
Sierpinski Triangles –
Used with permission from the author Cynthia Lanius http://math.rice.edu/~lanius/fractals/
8.1
Use triangle grid paper to do this activity.
[NOTE If you do not want to cut out the inside triangles, then colour the dark ones and leave the “cut out”
triangles white.]
Step One
Draw an equilateral triangle with sides of 2 triangle lengths each.
Connect the midpoints of each side.
How many equilateral triangles
do you now have?
Cut out the triangle
in the center.
Step Two
Draw another equilateral triangle with sides of 4 triangle lengths each.
Connect the midpoints of the sides and cut out the triangle
in the centre as before.
Notice the three small triangles that also need to be cut out in each of the
three triangles on each corner - three more holes.
Step Three
Draw an equilateral triangle with sides of 8 triangle lengths each.
Follow the same procedure as before, making sure to follow
the cutting pattern.
Step Four
For this one, you'll need a larger piece of paper, or cut smaller triangles.
Follow the above pattern and complete the fourth stage of the
Sierpinski Triangle.
Use your artistic creativity and shade the triangles in
interesting colour patterns.
Does your figure look like this one? Then you are correct!
8.
Going even further!
Step Five
1. Look at the triangle you made in Step One. What fraction of the triangle did you NOT cut
out?
2. What fraction of the triangle in Step Two is NOT cut out?
3. What fraction did you NOT cut out in the Step Three triangle?
4. Do you see a pattern here? Use the pattern to predict the fraction of the triangle you would
NOT cut out in the Step Four Triangle. Confirm your prediction and explain.
5. CHALLENGE: Develop a formula so that you could calculate the fraction of the area which is
NOT cut out for any step.
6. Write the fractions in the above questions in order from least to greatest. Write a statement
about how their order connects to the cutting out process.
7. Find another interesting pattern in the fractal called the Sierpinski Triangle.
Write a paragraph describing this pattern.
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