Lesson 33
Symmetry
4.G.A.3
Introduction
Use What You Know
You have learned about shapes and lines. Now you will learn about a line with a
particular purpose called a line of symmetry. Take a look at this problem.
Each of the figures below has a dashed line drawn across it. Imagine folding each
shape along the dotted line. If the two parts would fit exactly on top of each other
when the shape is folded, draw a star on that shape.
a. Trace the square, including the dashed line. Cut out the square and fold it on the
dashed line. Do the two parts fit exactly on top of each other? b.Now trace and cut out the rectangle. Make sure you include the dashed line. When
you fold on the dashed line, do the two parts fit exactly on top of each other? Why
or why not?
c. Which other shape(s) are divided into two parts that fit exactly on top of each other?
d.Which other shape(s) are divided into two parts that do not fit exactly on top of
each other? e. Describe what parts that fit exactly on top of each other look like and what parts
that don’t fit look like. 356
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Find Out More
In the problem on the previous page, you tested folding each figure along a line to
see if the two parts fit exactly on top of each other. When you can fold a shape on a
line and the parts line up with each other, the line is called a line of symmetry. Lines
of symmetry were shown on the square, the isosceles triangle, and the circle.
Shapes can have more than one line of symmetry.
All of the lines of symmetry for a square are shown on the square
at the right. Notice that all 4 lines of symmetry pass through the
center point of the square.
This scalene triangle has no lines of symmetry. If you try drawing
lines in the triangle so the parts fit exactly on top of each other,
you’ll see you can’t.
To the right are two different quadrilaterals. The
rectangle has two lines of symmetry which go
through the center point. The other quadrilateral has
no lines of symmetry.
Reflect
1 Describe a shape in your classroom that has at least one line of symmetry.
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Lesson 33 Symmetry
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Lesson 33
Modeled and Guided Instruction
Learn About
Finding a Line of Symmetry
Read the problem below. Then explore different ways to find lines
of symmetry.
Which of these shapes has at least one line of symmetry?
Model It You can trace and cut out the shapes. Then try folding the
shapes in half.
The shaded parts show where one part is folded over the other part.
Picture It You can try drawing different lines in the shapes to find
lines of symmetry.
The lines drawn show all of the places you could fold each shape to make parts that
fit exactly on top of each other.
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Connect It Now you will describe the lines of symmetry you found to solve
the problem from the previous page.
2 Does the parallelogram have any lines of symmetry? Why or why not?
3 How many lines of symmetry are shown on the pentagon? 4 Does the pentagon have any other lines of symmetry? How do you know?
5 The hexagon has 6 equal sides, 6 equal angles, and 6 lines of symmetry.
Explain why you think this is true. 6 Explain how you can figure out whether or not a line that divides a shape into two
parts is a line of symmetry.
Try It Use what you just learned to solve this problem.
7 Circle the figure below that has the greater number of lines of symmetry.
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Lesson 33 Symmetry
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Lesson 33
Modeled and Guided Instruction
Learn About
Drawing a Line of Symmetry
Read the problem below. Then explore different ways to draw lines
of symmetry.
Draw all of the lines of symmetry for these shapes. How many lines of symmetry
does each shape have?
Picture It You can use drawings to help draw lines of symmetry.
The top and bottom of the oval match,
so this shows a line of symmetry.
Another line can be drawn so that
the left and right sides match.
Picture It You can imagine folding the shape in different ways to draw lines
of symmetry.
Look at the plus sign. The lines show everywhere it could be folded to form parts that
fit on top of each other.
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Lesson 33 Symmetry
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Connect It Now you will describe and draw lines of symmetry to solve the
problem from the previous page.
8
On a separate piece of paper, trace the oval and show all lines of symmetry. What
do you notice about the place where all lines of symmetry cross?
9
Now trace the plus sign and show all lines of symmetry. Where do the lines cross?
How does this compare with the oval? 10 How many lines of symmetry does the oval have? 11 How many lines of symmetry does the plus sign have? 12 Explain how you can decide when you have found all of the lines of symmetry in
a figure. Try It Use what you just learned to solve this problem.
13 Draw all of the lines of symmetry on the hexagon below. All the sides of the figure
have the same length.
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Lesson 33 Symmetry
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Lesson 33
Practice
Guided Practice
Finding and Drawing Lines of Symmetry
Study the example below. Then solve problems 14–16.
Example
The student thought
about folding the figures
to decide where the lines
of symmetry are!
Which of the figures below has fewer lines of symmetry?
Look at how you could explain your work.
The square has lines of symmetry connecting the corners and
connecting both pairs of opposite sides.
Pair/Share
The rectangle only has lines of symmetry connecting opposite
sides, not opposite corners. So the rectangle has fewer lines
of symmetry.
Solution rectangle
Why do you think
that squares and
rectangles have
different numbers
of lines of symmetry?
14 Name a kind of triangle that has a line of symmetry. Name
another kind of triangle that doesn’t have a line of symmetry.
Show your work.
What are the kinds of
triangles that are named
by their sides?
Pair/Share
What’s different about
the triangle that has a
line of symmetry and
the one that doesn’t
have a line of
symmetry?
Solution 362
Lesson 33 Symmetry
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15 Draw all of the lines of symmetry on the figure. How many lines of
symmetry are there?
What point in a figure
do all lines of symmetry
pass through?
Pair/Share
Draw the figure on a
piece of grid paper
and cut it out to check
the lines of symmetry.
Solution 16 Which figure shows the correct line(s) of symmetry? Circle the
letter of the correct answer.
Imagine folding the
figures in half along the
lines.
A
C
B
D
Michael chose D as the correct answer. How did he get
that answer?
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Pair/Share
Discuss why the lines
of symmetry that are
incorrect don’t work.
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Lesson 33
Practice
Independent Practice
Finding and Drawing Lines of Symmetry
Solve the problems.
1 Which shape always has 4 lines of symmetry?
A an isosceles triangle
B a rhombus
C a square
D a rectangle
2 Each shape below has side lengths labeled in units. Determine the number of lines of
symmetry for each shape. Draw each shape in the correct box. Some boxes may have
more than one shape. Some boxes may not have any shapes.
3
6
5
2
2
7
7
5
3
4
Number of
Lines of
Symmetry
0
Exactly
1
4
4
4
3
3
3
3
5
5
4
3
Exactly
2
Exactly
3
Exactly
4
More
than
4
Shape
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3 Part of a figure is shaded on the grid below. Complete the figure by shading squares.
Lines r and s are lines of symmetry for the completed figure.
r
s
4 Draw all of the lines of symmetry on the figure below. Then draw a different
quadrilateral that has more lines of symmetry than this figure. Show the lines
of symmetry.
Explain your work.
Self Check Go back and see what you can check off on the Self Check on page 331.
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