Download Lesson 8 - Engaging Students

Math 5
Unit 3
Lesson 8
Parallelograms
Folk Music
Zach’s family loves to play in a folk music band. They have many types of
instruments that they play. Zach found that in between some of his frets
on this guitar, there are decorative parallelograms. The frets
are the bars that the strings
press against.
Reflection
Can you find examples of parallelograms in
your environment?
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Math 5
Unit 3
Lesson 8: Parallelograms
Objectives for this Lesson
In this lesson you will explore the following concepts:
• Describe the attributes of a parallelogram
• Relate rectangle attributes to parallelogram attributes
• Given the measures of the sides find the perimeter
Go online to watch the Notepad Tutor about Classifying Polygons.
Sides of a Parallelogram
You should remember that a quadrilateral is a polygon with four sides.
Here are some quadrilaterals:
As you can see by the figures here, quadrilateral can mean some very
different things. In order to be more descriptive we can limit the name
quadrilateral with other names.
A parallelogram is a quadrilateral. When you use the name parallelogram
you have narrowed down the first set of quadrilaterals to the following:
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Lesson 8: Parallelograms
Look at the sides of the parallelograms and make observations. There are
two attributes of the sides of a parallelogram. Attributes are rules for
defining the parallelogram.
The sides of a parallelogram have these attributes:
Opposite sides are parallel
Opposite sides are congruent
Be careful! In this figure there is one pair of opposite sides that
are parallel:
Why isn’t this figure a parallelogram?
The blue sides are parallel, but the other attribute is ignored.
The opposite sides are not congruent.
Example 1
Which of these shapes have two pairs of opposite sides that are parallel?
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Lesson 8: Parallelograms
The answer is:
and
The other two have one pair of sides that are parallel but those sides are
not congruent.
Example 2
Measure the sides of the parallelogram in centimetres. Are opposite
sides congruent?
Use a centimetre ruler. Line up the 0 mark with one endpoint and
measure to the other endpoint:
This side is 5.1 centimetres. Measure the opposite side. It also measures
5.1 centimetres. This pair of sides is congruent.
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Lesson 8: Parallelograms
Now measure a short side:
This side is 2.2 centimetres. Measure the opposite side. It also measures
2.2 centimetres.
The answer is:
One pair of opposite sides measure 5.1 cm and are congruent.
The other pair of opposite sides measure 2.2 cm and are congruent.
Angles of a Parallelogram
The angles of a parallelogram follow the attribute:
Opposite Angles are Congruent
So what are opposite angles? In a quadrilateral there are two ways to
describe a pair of angles. Pairs of angles are either opposite angles
or consecutive angles.
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Unit 3
Lesson 8: Parallelograms
Opposite angles are found opposite of one another inside the figure:
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gle
An
te
i
s
o
pp
O
Consecutive angles are connected inside the figure:
Consecutive Angles
Example 3
Use the numbers that mark the angles to complete the statements.
1
4
2
3
A. A
ngle 1 is opposite angle _______.
Angle 1 is opposite angle 3. This is the only answer for this statement.
B. A
ngle 1 and Angle _______ are consecutive angles.
Angle 1 and Angle 2 OR 4 are consecutive angles, depending on
whether you go down or across from Angle 1. Either answer is correct.
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Lesson 8: Parallelograms
C.Angle 2 is opposite angle _______.
Angle 2 is opposite angle 4. This is the only answer for this statement.
D.Name two angles that are consecutive angles to Angle 3.
Angle 2 and Angle 4 are consecutive angles to Angle 3.
Reflection
How can you tell if the angles are congruent?
Let’s Explore
Exploration 1: Angle Attributes of Parallelograms
Materials: Unit 3, Lesson 8, Exploration 1 page from your Workbook, Scissors, Pencil,
Page of parallelograms from the back of this Unit in your Workbook
1. C
ut out the parallelograms on the page found at the back of this
Unit in your Workbook.
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Lesson 8: Parallelograms
2. Rip corners 2 and 4 off of Parallelogram A as shown:
1
2
A
4
3
3. Now overlap Angles 2 and 4 so that they match up – or not.
Do the edges of the angles overlap? If so, they are congruent.
4. Repeat the test on Angles 1 and 3.
5. Repeat the test on Parallelograms B and C.
6. Are opposite angles always congruent?
7. A
re there any other observations you can make about angles
in a parallelogram?
Are These Parallelograms?
Here is a group of parallelograms:
2
2
6
2
2
3
1
1
6
4
3
5
3
3
5
4
All of the sides of the figures are marked with their measures. This should
tell you that they all have opposite sides that are congruent. You should
also be able to tell that they are parallel.
All of these figures are parallelograms, but three have special names.
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Lesson 8: Parallelograms
There are special parallelograms that have more attributes:
Rectangle
ALL angles are right angles
Rhombus
All sides are congruent
Square
ALL angles are right angles
All sides are congruent
These figures have the attributes of parallelograms. They also have these
extra attributes that make them special.
Example 4
Nina and her father have made a garden in their backyard. She says that
the garden is shaped like a parallelogram.
Cameron says that it
is shaped like a
rectangle. He shows
Nina that the corners
are all right angles by
putting his clipboard
in each one.
Who is right and why?
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Lesson 8: Parallelograms
The garden has all the attributes of a parallelogram:
• opposite sides are parallel
• opposite sides are congruent
• opposite angles are congruent
This means that Nina is right! It is a parallelogram.
The angles in the corners are all right angles. This means that Cameron
is right! It is also a rectangle.
How often are two friends right when they disagree? This may be a first!
Let’s Practice
• Turn in your Workbook to Unit 3, Lesson 8 and complete 1 to 12.
Perimeter
Nina wants to put a fence around her garden. She will need to know how
much fence material to buy. She measures two sides of her garden:
12 m
16 m
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Lesson 8: Parallelograms
Can Nina find the perimeter of her garden with only these
two measurements?
The perimeter of parallelograms is found by finding the distance around.
You probably remember that this is done by simply adding up the sides.
10
4
4
10
Perimeter = 4 + 10 + 4 +10 = 28
If you know that the figure is a parallelogram then you know that the
opposite sides are congruent. You can use that attribute to find the
missing sides.
5
Opposite sides are
Congruent
8
5
8
8
5
Now let’s try one.
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Lesson 8: Parallelograms
Example 5
Find the perimeter of the parallelogram.
14
12
Find the missing sides using the attribute “opposite sides are congruent.”
In this case that means that they are the same measure:
14
12
12
14
Add up all of the side lengths to get the perimeter: 12 + 14 + 12 + 14 = 52
The perimeter is: 52
Using the same method you can find the perimeter of Nina’s garden.
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Example 6
Find the perimeter of Nina’s garden.
12 m
16 m
Nina was so smart when she took the measurements! She measured one
short side and one long side. She can use the attribute “opposite sides
are congruent” to find the missing sides:
16 m
12 m
12 m
16 m
Now she can find the perimeter by adding all four sides:
12 + 16 + 12 + 16 = 56 metres
Let’s Practice
• Turn in your Workbook to Unit 3, Lesson 8 and complete 13 to 18.
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Lesson 8: Parallelograms