Download 8th Grade Math 2D Geometry: Transformations Transformations

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New Jersey Center for Teaching and Learning
8th Grade Math
Progressive Mathematics Initiative
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2D Geometry:
Transformations
2013-12-09
www.njctl.org
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Table of Contents
Click on a topic to
go to that section
· Transformations
· Translations
Transformations
· Rotations
· Reflections
·
·
·
·
Dilations
Symmetry
Congruence & Similarity
Special Pairs of Angles
Return to
Table of
Contents
Common Core Standards: 8.G.1, 8.G.2, 8.G.3, 8.G.4, 8.G.5
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Any time you move, shrink, or enlarge a figure you make a
transformation. If the figure you are moving (pre-image) is
labeled with letters A, B, and C, you can label the points on
the transformed image (image) with the same letters and the
prime sign.
Pull
image
pre-image
C C'
B
B'
A'
A
Triangle ABC is the pre-image to the reflected image triangle XYZ
for transformation shown
B
The image can also be labeled with new letters as shown below.
Y
X
A
image
pre-image
C Z
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There are four types of transformations in this unit:
There are four types of transformations in this unit:
·
·
·
·
Translations
Rotations
Reflections
Dilations
·
·
·
·
The first three transformations preserve the size and shape of the
figure. They will be congruent. Congruent figures are same size
and same shape.
In other words:
If your pre-image is a trapezoid, your image is a congruent
trapezoid.
If your pre-image is an angle, your image is an angle with the same
measure.
If your pre-image contains parallel lines, your image contains
parallel lines.
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Translations
Rotations
Reflections
Dilations
The first three transformations preserve the size and shape of the
figure.
In other words:
If your pre-image is a trapezoid, your image is a congruent
trapezoid.
If your pre-image is an angle, your image is an angle with the same
measure.
If your pre-image contains parallel lines, your image contains
parallel lines.
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Translations
Return to
Table of
Contents
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A translation is a slide that moves a figure to a different
position (left, right, up or down) without changing its
size or shape and without flipping or turning it.
You can use a slide arrow to
show the direction and
distance of the movement.
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This shows a translation of pre-image ABC to image
A'B'C'. Each point in the pre-image was moved
right 7 and up 4.
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To complete a translation, move each point of the pre-image and
label the new point.
Click for web page
B'
A'
PULL
Example: Move the figure left 2 units and up 5 units.
What are the coordinates of the pre-image and image?
C'
D'
A
D
B
C
Are the line segments in the pre-image and image the same
length? In other words, was the size of the figure preserved?
Both the pre-image and image are congruent.
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Translate pre-image ABCD 4 right and 1 down.
What are the coordinates of the image and pre-image?
Translate pre-image ABC 2 left and 6 down.
What are the coordinates of the image and pre-image?
A
PULL
A
PULL
C
B
D
B
C
Are the line segments in the pre-image and image the same
length? In other words, was the size of the figure preserved?
Both the pre-image and image are congruent.
Are the line segments in the pre-image and image the same
length? In other words, was the size of the figure preserved?
Both the pre-image and image are congruent.
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A rule can be written to describe translations on the
coordinate plane. Look at the following rules and
coordinates to see if you can find a pattern.
Translate pre-image ABCD 5 left and 3 up.
What are the coordinates of the image and pre-image?
PULL
A
B
C
D
Are the line segments in the pre-image and image the same
length? In other words, was the size of the figure preserved?
Both the pre-image and image are congruent.
2 Left and 5 Up
A (3,-1) A' (1,4)
B (8,-1) B' (6,4)
C (7,-3) C' (5,2)
D (2, -4) D' (0,1)
2 Left and 6 Down
A (-2,7) A' (-4,1)
B (-3,1) B' (-5,-5)
C (-6,3) C' (-8,-3)
5 Left and 3 Up
A (3,2) A' (-2,5)
B (7,1) B' (2,4)
C (4,0) C' (-1,3)
D (2,-2) D' (-3,1)
4 Right and 1 Down
A (-5,4) A' (-1,3)
B (-1,2) B' (3,1)
C (-4,-2) C' (0,-3)
D (-6, 1) D' (-2,0)
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Translating left/right changes the x-coordinate.
Translating left/right changes the x-coordinate.
· Left subtracts from the x-coordinate
Translating up/down changes the y-coordinate.
2 Left and 5 Up
A (3,-1) A' (1,4)
B (8,-1) B' (6,4)
C (7,-3) C' (5,2)
D (2, -4) D' (0,1)
2 Left and 6 Down
A (-2,7) A' (-4,1)
B (-3,1) B' (-5,-5)
C (-6,3) C' (-8,-3)
5 Left and 3 Up
A (3,2) A' (-2,5)
B (7,1) B' (2,4)
C (4,0) C' (-1,3)
D (2,-2) D' (-3,1)
4 Right and 1 Down
A (-5,4) A' (-1,3)
B (-1,2) B' (3,1)
C (-4,-2) C' (0,-3)
D (-6, 1) D' (-2,0)
· Right adds to the x-coordinate
Translating up/down changes the y-coordinate.
· Down subtracts from the y-coordinate
· Up adds to the y-coordinate
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Write a rule for each translation.
A rule can be written to describe translations on the coordinate
plane.
2 units Left …
x-coordinate - 2
y-coordinate
stays
click to reveal
rule = (x - 2, y)
5 units Right & 3 units Down…
x-coordinate + 5
y-coordinate
-3
click to reveal
rule = (x + 5, y - 3)
2 Left and 5 Up
A (3,-1) A' (1,4)
B (8,-1) B' (6,4)
C (7,-3) C' (5,2)
D (2, -4) D' (0,1)
2 Left and 6 Down
A (-2,7) A' (-4,1)
B (-3,1) B' (-5,-5)
C (-6,3) C' (-8,-3)
click
reveal
(x,
y) to(x-2,
y+5)
click
reveal
(x,
y) to(x-2,
y-6)
5 Left and 3 Up
A (3,2) A' (-2,5)
B (7,1) B' (2,4)
C (4,0) C' (-1,3)
D (2,-2) D' (-3,1)
4 Right and 1 Down
A (-5,4) A' (-1,3)
B (-1,2) B' (3,1)
C (-4,-2) C' (0,-3)
D (-6, 1) D' (-2,0)
(x,click
y) to
(x-5,
y+3)
reveal
reveal
(x,click
y) to
(x+4,
y-1)
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1
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What rule describes the translation shown?
A
(x,y)
E'
(x - 4, y - 6)
B
(x,y)
(x - 6, y - 4)
C
(x,y)
(x + 6, y + 4)
D
(x,y)
(x + 4, y + 6)
2
D'
D
F'
E
F
G'
Pull
What rule describes the translation shown?
A
(x,y)
(x, y - 9)
B
(x,y)
(x, y - 3)
C
(x,y)
(x - 9, y)
D
(x,y)
(x - 3, y)
D
G
D'
E
F
G
E'
F'
G'
Pull
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3
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What rule describes the translation shown?
A
(x,y)
(x + 8, y - 5)
B
(x,y)
(x - 5, y - 1)
C
(x,y)
(x + 5, y - 8)
D
(x,y)
(x - 8, y + 5)
D'
4
Pull
E'
F'
G'
E
D
What rule describes the translation shown?
A
(x,y)
(x - 3, y + 2)
B
(x,y)
(x + 3, y - 2)
C
(x,y)
(x + 2, y - 3)
D
(x,y)
(x - 2, y + 3)
D
Pull
E
D'
F
E'
F'
G
F
G'
G
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5
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What rule describes the translation shown?
A (x,y)
(x - 3, y + 2)
B (x,y)
(x + 3, y - 2)
C (x,y)
(x + 2, y - 3)
D (x,y)
(x - 2, y + 3)
Pull
E'
D'
F'
E
D
F
Rotations
G'
G
Return to
Table of
Contents
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A rotation (turn) moves a figure around a point. This point
can be on the figure or it can be some other point. This point
is called the point of rotation.
P
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When you rotate a figure, you can describe the rotation by
giving the direction (clockwise or counterclockwise) and
the angle that the figure is rotated around the point of
rotation. Rotations are counterclockwise unless you are
told otherwise. Describe each of the rotations.
Rotation
Hint
A
The person's finger is the
point of rotation for each
figure.
B
This figure is rotated
Click
for answer
180
degrees
clockwise
about point B.
This figure is rotated
Click
for answer
90 degrees
counterclockwise
about point A.
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How is this figure
rotated about the
origin?
In a coordinate
plane, each
quadrant
represents
B
A
B'
D
A'
C
C'
D'
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In order to
determine the
angle, draw
two rays (one
from the point
of rotation to
pre-image
point, the
other from the
point of
rotation to the
image point).
Measure this
angle.
The following descriptions describe the same rotation.
What do you notice?
Can you give your own example?
This figure is rotated 270 degrees clockwise
Click
about the origin
orto
90Reveal
degrees
counterclockwise about the origin.
Check to see if the pre-image and image are congruent.
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The sum of the two rotations (clockwise and counterclockwise)
is 360 degrees.
If you have one rotation, you can calculate the other by
subtracting from 360.
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6
How is this figure rotated about point A? (Choose more
than one answer.)
C
A
clockwise
B
counterclockwise
C
90 degrees
D
180 degrees
E
270 degrees
D
C'
B'
E
D'
B
A, A'
E'
Check to see if the pre-image and image are congruent.
Pull
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7
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How is this figure rotated about point the origin? (Choose
more than one answer.)
A
clockwise
B
counterclockwise
C
90 degrees
D
180 degrees
E
270 degrees
A
B
D
C
Pull
C'
A'
Pull
B'
A'
C
C'
D'
When rotated
counter-clockwise, the x-coordinate is the
opposite of the pre-image y-coordinate and the y-coordinate is
the same as the pre-image of the x-coordinate. In other words:
Click to Reveal
(x, y)
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Write the
coordinates for the
pre-image and
image.
D
What do you
notice?
Check to see if the pre-image and image are congruent.
What happens to
the coordinates in
a half-turn?
B
A
Write the
coordinates for the
pre-image and
image.
D'
B'
Now let's look at
the same figure
and see what
happens to the
coordinates when
we rotate a figure.
(-y, x)
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Can you summarize what happens to the coordinates during
a rotation?
A
D
What do you
notice?
B
Pull
Counterclockwise:
(x, y)to Reveal
(-y, x)
Click
C
C'
B'
D'
Half-turn:
(x, y) to Reveal
(-x, -y)
Click
A'
Clockwise:
(x, y)
(y, -x)
Click to Reveal
When rotated a half-turn, the x-coordinate is the opposite of the
pre-image x-coordinate and the y-coordinate is the opposite of the
pre-image of the y-coordinate.
other words:
Click toInReveal
(x, y)
(-x, -y)
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9
What are the new coordinates of a point A (5, -6) after a
rotation clockwise?
8
A (-6, -5)
B (6, -5)
Pull
What are the new coordinates of a point S (-8, -1) after a
rotation counterclockwise?
A (-1, -8)
B (1, -8)
C
(-5, 6)
C
(-1, 8)
D
(5, -6)
D
(8, 1)
Pull
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What are the new coordinates of a point H (-5, 4) after a
rotation counterclockwise?
10
A (-5, -4)
11
Pull
B (5, -4)
What are the new coordinates of a point R (-4, -2) after a
rotation clockwise?
A (2, -4)
Pull
B (-2, 4)
C
(4, -5)
C
(2, 4)
D
(-4, 5)
D
(-4, 2)
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12 What are the new coordinates of a point Y (9, -12) after a
half-turn?
A (-12, 9)
Pull
B (-9,12)
C
(-12, -9)
D
(9,12)
Reflections
Return to
Table of
Contents
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Examples
A reflection (flip) creates a mirror image of a figure.
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A reflection is a flip because the figure is flipped over a
line. Each point in the image is the same distance from
the line as the original point.
t
A and A' are both 6 units from line t.
B and B' are both 6 units from line t.
C and C' are both 3 units from line t.
A
A'
B
C
y
A
B
D
C
Each vertex in
ABC is the same
distance from line t as the vertices in
A'B'C'.
B'
C'
Reflect the figure across the y-axis.
Check to see if the pre-image and image are congruent.
x
Check to see if the pre-image and image are congruent.
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What do you notice about the coordinates when you reflect
across the y-axis?
y
A
D
B
C
Tap box for coordinates
B'
A (-6, 5)
B (-4, 5)
C (-4, 1)
D (-6, 3)
A'
D'
C'
x
A' (6, 5)
B' (4, 5)
C' (4, 1)
D' (6, 3)
When you reflect across
the y-axis, the x-coordinate
becomes the opposite.
So (x, y)
(-x, y) when
you reflect across the yaxis.
Check to see if the pre-image and image are congruent.
What do you predict about the coordinates when you
reflect across the x-axis?
Tap box for coordinates
y
A
D
A (-6, 5)
B (-4, 5)
C (-4, 1)
D (-6, 3)
B
C
D'
A'
B'
So (x, y)
(x, -y) when
you reflect across the xaxis.
Check to see if the pre-image and image are congruent.
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Reflect the figure across the y-axis then the x-axis.
Click to see each reflection.
y
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Reflect the figure across the y-axis.
Click to see reflection.
y
A
A
B
B
C
D
When you reflect across
the x-axis, the y-coordinate
becomes the opposite.
x
C'
A' (-6, -5)
B' (-4, -5)
C' (-4, -1)
D' (-6, -3)
C
x
Check to see if the pre-image and image are congruent.
F
D
E
x
Check to see if the pre-image and image are congruent.
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Reflect the figure across the line x = -2.
Reflect the figure across the line y = x.
y
y
B
C
A
E
x
D
Check to see if the pre-image and image are congruent.
A
B
D
C
Check to see if the pre-image and image are congruent.
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13 The reflection below represents a reflection across:
14 The reflection below represents a reflection across:
A
the x axis
C
the x axis, then the y axis
A
the x axis
B
the y axis
D
the y axis, then the x axis
B
the y axis
y
y
B
B'
A
Pull
A
the x axis, then the y axis
D
the y axis, then the y axis
D
C
B
C
x
C'
C'
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Which of the following represents a single reflection
of Figure 1?
15
16
Which of the following describes the movement below?
A
Figure 1
D
A'
Check to see if the pre-image and image are congruent.
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C
x
B'
D'
Check to see if the pre-image and image are congruent.
B
C
Pull
A'
A
x
Pull
reflection
B
rotation, 90 clockwise
C
slide
D
rotation, 180 clockwise
Pull
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17 Describe the reflection below:
18 Describe the reflection below:
A
across the line y = x
C
across the line y = -3
A
across the line y = x
C
across the line y = -3
B
across the y axis
D
across the x axis
B
across the x axis
D
across the line x = 4
y
Pull
D'
C'
E'
B'
A'
Pull
y
A'
A
B
C
B
x
A
E
B'
C C'
x
D
Check to see if the pre-image and image are congruent.
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Check to see if the pre-image and image are congruent.
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Dilations
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Table of
Contents
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The scale factor is the ratio of sides:
A dilationis a transformation in which a figure is enlarged or
reduced around a center point using a scale factor = 0.
The center point is not altered.
When the scale factor of a dilation is greater than 1, the
dilation is an enlargement .
When the scale factor of a dilation is less than 1, the dilation
is a reduction.
When the scale factor is |1|, the dilation is an identity.
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Example.
What happened to the coordinates with a scale factor of 2?
If the pre-image is dotted and the image is solid, what type of
dilation is this? What is the scale factor of the dilation?
y
y
x
A (0, 1)
B (3, 2)
C (4, 0)
D (1, 0)
B'
This is an enlargement.
Scale Factor is 2.
A'
A
to reveal
base length Click
of image
base length of pre-image
B
D D' C
6
3 = 2
C'
x
A' (0, 2)
B' (6, 4)
C' (8, 0)
D' (2, 0)
The coordinates were all
Click to reveal
multiplied
by 2.
The center for this dilation
was the origin (0,0).
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20What
are the coordinates of a point S (3, -2)
after a dilation with a scale factor of 4 about
the origin?
19What
is the scale factor for the image
shown below? The pre-image is dotted and
y
the image is solid.
A 2
A (12, -8)
B 3
Pull
C -3
B (-12, -8)
C (-12, 8)
D 4
D (-3/4, 1/2)
x
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22What
are the coordinates of a point X (4, -8)
after a dilation with a scale factor of 0.5?
21What
are the coordinates of a point Y (-2, 5)
after a dilation with a scale factor of 2.5?
A (-8, 16)
A (-0.8, 2)
B (-5, 12.5)
Pull
Pull
B (8, -16)
C (0.8, -2)
C (-2, 4)
D (5, -12.5)
D (2, -4)
Pull
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The coordinates of a point change as follows
during a dilation:
(-6, 3)
(-2, 1)
23
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24The
coordinates of a point change as follows
during a dilation:
(4, -9)
(16, -36)
Pull
What is the scale factor?
What is the scale factor?
A 3
A 4
B -3
B -4
C 1/3
C 1/4
D -1/3
D -1/4
Pull
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26
25The
coordinates of a point change as follows
during a dilation:
(5, -2)
(17.5, -7)
Which of the following figures represents a rotation?
(and could not have been achieved only using a reflection)
A Figure A
B Figure B
Pull
What is the scale factor?
Pull
A 3
B -3.75
C Figure C
C -3.5
D Figure D
D 3.5
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27Which
Slide 78 / 168
of the following figures represents a reflection?
A
Figure A
B
28
Figure B
Which of the following figures represents a dilation?
B Figure B
A Figure A
Pull
Pull
C
Figure C
D
Figure D
C
Figure C
D
Figure D
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29
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Which of the following figures represents a translation?
B Figure B
A Figure A
Pull
C
Figure C
D
Symmetry
Figure D
Return to
Table of
Contents
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Symmetry
A line of symmetry divides a figure into two parts that match
each other exactly when you fold along the dotted line. Draw the
lines of symmetry for each figure below if they exist.
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Which of these figures have symmetry?
Draw the lines of symmetry.
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Do these images have symmetry? Where?
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We think that our faces are symmetrical, but most faces
are asymmetrical (not symmetrical). Here are a few
pictures of people if their faces were symmetrical.
Will Smith with a symmetrical face.
Click the picture below to learn how to make
your own face symmetrical.
Marilyn Monroe
with a
symmetrical
face.
Tina Fey
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Slide 88 / 168
Rotational symmetry is when a figure can be rotated
around a point onto itself in less than a 360 turn. Rotate
these figures. What degree of rotational symmetry do
each of these figures have?
Rotational symmetry is when a figure can be rotated
around a point onto itself in less than a 360 turn.
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30
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How many lines of symmetry does this figure have?
31Which
A 3
B 6
C 5
D 4
Pull
A
figure's dotted line shows a line of symmetry?
B
C
D
Pull
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32
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Which of the objects does not have rotational symmetry?
A
Pull
B
C
Congruence &
Similarity
D
Return to
Table of
Contents
Rotational symmetry is when a figure can be rotated
around a point onto itself in less than a 360 turn.
Click for hint.
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Congruence and Similarity
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Similar shapes have the same shape, congruent angles and
proportional sides.
Congruent shapes have the same size and shape.
2 figures are congruent if the second figure can be
obtained from the first by a series of translations,
reflections, and/or rotations.
Pull
2 figures are similar if the second figure can be obtained
from the first by a series of translations, reflections,
rotations and/or dilations.
Remember - translations, reflections and rotations
preserve image size and shape.
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Example
Click for web page
What would the measure of angle j have to be in order for the
figures below to be similar?
j
180 - 112 - 33 = 35
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Example
33Which
Are the two triangles below similar? Explain your reasoning?
pair of shapes is similar but not congruent?
A
Pull
B
C
Yes, the triangles have congruent
angles and areClick
therefore similar.
D
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34Which
Slide 100 / 168
Which of the following terms best describes the pair of
figures?
pair of shapes is similar but not congruent?
35
A
B
Pull
A
congruent
B
similar
C
neither congruent nor similar
Pull
C
D
Slide 101 / 168
Slide 102 / 168
Which of the following terms best describes the pair of figures?
37
36Which
of the following terms best describes the pair of
figures?
A
congruent
A
congruent
B
similar
B
similar
C
neither congruent nor similar
C
neither congruent nor similar
Pull
Pull
Slide 103 / 168
Slide 104 / 168
Determine if the two figures are congruent, similar or
neither.
Determine if the two figures are congruent, similar or
neither.
Be able to explain how one figure was obtained from the
other through a series of translations, rotations, reflections
and/or dilations. The pre-image is dotted, the image is solid.
Be able to explain how one figure was obtained from the
other through a series of translations, rotations, reflections
and/or dilations. The pre-image is dotted, the image is solid.
Pull
Pull
Slide 105 / 168
Slide 106 / 168
Determine if the two figures are congruent, similar or
neither.
Determine if the two figures are congruent, similar or
neither.
Be able to explain how one figure was obtained from the
other through a series of translations, rotations, reflections
and/or dilations. The pre-image is dotted, the image is solid.
Be able to explain how one figure was obtained from the
other through a series of translations, rotations, reflections
and/or dilations. The pre-image is dotted, the image is solid.
Pull
Pull
Slide 107 / 168
Slide 108 / 168
Recall:
·
Complementary Angles are two angles with a sum of 90 degrees.
These two angles are complementary
angles because their sum is 90.
Special Pairs of
Angles
Notice that they form a right angle
when placed together.
· Supplementary Angles are two angles with a sum of 180 degrees.
Return to
Table of
Contents
These two angles are supplementary
angles because their sum is 180.
Notice that they form a straight angle
when placed together.
Slide 109 / 168
Slide 110 / 168
Transformations
Vertical Angles are two angles that are opposite each
other when two lines intersect.
a
b
d
Vertical Angles can further be explained using the transformation
of reflection.
b
c
c
a
d
x
In this example, the vertical angles are:
Line x cuts angles b and d in half.
Vertical angles have the same measurement.
So:
When angle a is reflected over line x, it forms angle c.
When angle c is reflected over line x, it forms angle a.
Slide 111 / 168
Slide 112 / 168
Transformations Continued
Using what you know about complementary,
supplementary and vertical angles, find the measure of
the missing angles.
y
b
c
a
d
b
c
a
Line y cuts angles a and c in half.
By Vertical Angles:
When angle b is reflected over line y, it forms angle d.
Click
When angle d is reflected over line y, it forms angle b.
Slide 113 / 168
38Are
By Supplementary Angles:
Click
Slide 114 / 168
angles 2 and 4 vertical angles?
39
Yes
Are angles 2 and 3 vertical angles?
Yes
No
No
1
2
4
3
Pull
3
2
4
1
Pull
Slide 115 / 168
40
Slide 116 / 168
41
If angle 1 is 60 degrees, what is the measure of
angle 3? You must be able to explain why.
A 30 o
B 60
C 120
A 30 o
Pull
o
o
B 60 o
C 120
2
D 15 o
1
4
If angle 1 is 60 degrees, what is the measure of
angle 2? You must be able to explain why.
Slide 117 / 168
3
Adjacent or Not Adjacent?
You Decide!
A
a
is adjacent to
b
How do you know?
· They have a common side (ray
)
· They have a common vertex (point B)
a
a
click
to reveal
Adjacent
b
b
clickAdjacent
to reveal
Not
click
to reveal
Not
Adjacent
D
B
Slide 119 / 168
42
4
Slide 118 / 168
Adjacent Angles are two angles that are next to each
other and have a common ray between them. This
means that they are on the same plane and they share
no internal points.
C
1
D 15 o
3
Pull
2
o
Slide 120 / 168
43
Which two angles are adjacent to each other?
A
1 and 4
B
2 and 4
Which two angles are adjacent to each other?
A
3 and 6
B
5 and 4
Pull
Pull
1
4
2
5
6
3
2
4
1
3 6
5
Slide 121 / 168
Slide 122 / 168
Recall From 3rd Grade
A transversal is a line that cuts across two or more (usually
parallel) lines.
Shapes and Perimeters
Parallel lines are a set of two lines that do not intersect (touch).
A
P
E
Q
F
A
R
B
Interactive Activity-Click Here
Slide 123 / 168
Slide 124 / 168
44
T ra
nsv
e rs
al
Corresponding Angles are on the same side of the transversal
and on the same side of the given lines.
In this diagram the
corresponding
angles are:
a
c
d
b
Which are pairs of corresponding angles?
A
2 and 6
B
3 and 7
C
1 and 8
Pull
1
3
5
f
e
g
7
2
4
6
8
h
Click
Slide 125 / 168
45
Slide 126 / 168
46
Which are pairs of corresponding angles?
A
2 and 6
B
3 and 1
C
Which are pairs of corresponding angles?
A 1 and 5
Pull
1 and 8
6
2
1
7
2
1
4
3
C 4 and 8
5
7
4
8
3
5
B 2 and 8
6
8
Pull
Slide 127 / 168
Slide 128 / 168
47
Which are pairs of corresponding
angles ?
A 2 and 4
5
4
C 7 and 8
2
In this diagram the
alternate interior
angles are:
a
c
d
b
m
6
3
7
D 1 and 3
Pull
1
8
B 6 and 5
Alternate Interior Angles are on opposite sides of the
transversal and on the inside of the given lines.
l
g
n
f
e
h
Click
Slide 129 / 168
Slide 130 / 168
Alternate Exterior Angles are on opposite sides of the
transversal and on the outside of the given lines.
l
Same Side Interior Angles are on same sides of the
transversal and on the inside of the given lines.
l
In this diagram the
same side interior
angles are:
a
c
g
In this diagram the
alternate exterior
angles are:
m
a
c
n
f
e
Click
d
b
g
Click
Slide 131 / 168
48
49
l
Pull
No
3
1
5
2
6
8
4
7
h
m
n
Which line is the
transversal?
Slide 132 / 168
Are angles 2 and 7 alternate exterior angles?
Yes
b
f
e
h
d
Are angles 3 and 6 alternate exterior angles?
l
Yes
Pull
No
3
1
m
5
n
2
6
8
4
7
m
n
Slide 133 / 168
50
Slide 134 / 168
Are angles 7 and 4 alternate exterior angles?
51
Which angle corresponds to angle 5?
l
Yes
No
3
5
2
6
C
m
2
6
Slide 135 / 168
52Which
53
Pull
B
C
3
1
D
5
What type of angles are
What type of angles are
7
4
l
Pull
B Alternate Exterior Angles
Same Side Interior
2
3
m
7
4
n
8
What type of angles are
and
?
A Alternate Interior Angles
5
Same Side Interior
2
6
8
4
1 3
7
C Corresponding Angles
m
1
D Vertical Angles
E
n
Pull
l
B Alternate Exterior Angles
D Vertical Angles
E
1
5
6
55
C Corresponding Angles
Pull
l
Slide 138 / 168
?
A Alternate Interior Angles
?
D Vertical Angles
n
8
and
and
C Corresponding Angles
m
Slide 137 / 168
54
n
B Alternate Exterior Angles
E
2
4
8
A Alternate Interior Angles
l
A
m
7
Slide 136 / 168
pair of angles are same side interior?
6
5
n
8
3
1
D
7
4
Pull
B
Pull
1
l
A
5
Same Side Interior
2
6
8
4
3
7
m
n
Slide 139 / 168
56
Slide 140 / 168
Are angles 5 and 2 alternate interior angles?
57
l
Yes
No
Pull
1 3
5
7
2
l
Yes
No
6
Are angles 5 and 7 alternate interior angles?
Pull
1
m
5
4
2
n
8
6
4
n
Slide 142 / 168
Are angles 7 and 2 alternate interior angles?
59
l
Yes
m
8
Slide 141 / 168
58
3
7
Are angles 3 and 6 alternate exterior angles?
l
Yes
No
No
3
1
5
2
6
Pull
m
7
5
n
4
2
6
8
Pull
3
1
m
7
4
n
8
Slide 143 / 168
Slide 144 / 168
Special Cases
Reflections
If parallel lines are cut by a transversal then:
n
· Corresponding Angles are congruent
1
5
· Alternate Interior Angles are congruent
2
· Same Side Interior Angles are supplementary
SO:
6
n
1
click
5
are supplementary
6
3
m
7
4
2
7
b
· Alternate Exterior Angles are congruent
are supplementary
a
m
3
8
These Special Cases can further be explained using the
transformations of reflections and translations
l
4
l
8
Line a cuts angles 3 and 5
in half.
Line b cuts angles 4 and 6
in half.
When angle 1 is reflected
over line a, it forms angle 7.
When angle 2 is reflected
over line b, it forms angle 8.
When angle 7 is reflected
over line a, it forms angle 1.
When angle 8 is reflected
over line b, it forms angle 2.
Slide 145 / 168
Slide 146 / 168
Reflections Continued
1
1
m
3
5
Translations
n
n
c
5
m
3
7
Line m is parallel to line l.
7
d
2
6
2
l
4
6
l
4
8
8
n
Line c cuts angles 1 and 7
in half.
Line d cuts angles 2 and 8
in half.
When angle 3 is reflected
over line c, it forms angle 5.
When angle 4 is reflected
over line d, it forms angle 6.
2
6
When angle 5 is reflected
over line c, it forms angle 3.
1
4
8
5
Slide 148 / 168
Translations Continued
If line m is then translated x units
left, all angles formed by lines m
and n will overlap with all angles
formed by lines l and n.
l
m
43
Given the measure of one angle, find the
measures of as many angles as possible.
Which angles are congruent to the given angle?
60
n
1
2
ml
7
When angle 6 is reflected
over line d, it forms angle 4.
Slide 147 / 168
56
3
If line m is translated y units
down, it will overlap with line l.
l
A <4, <5, <6
B <5, <7, <1
4
C <2
78
5
m
6
Pull
D <5, <1
n
12
56
43
2
The translations also work
if line l is translated y units
up and x units right.
l
m
78
1
Slide 149 / 168
Given the measure of one angle, find the
measures of as many angles as possible.
What are the measures of angles 4, 6, 2 and 8?
Given the measure of one angle, find the
measures of as many angles as possible.
Which angles are congruent to the given angle?
4
5
6
o
A <4
l
B <4, <5, <3
m
Pull
C 130
62
l
B 40 o
n
Slide 150 / 168
61
A 50 o
7
8
5
D <8
2
1
2
7
8
n
3
1
C <2
8
4
Pull
m
7
n
Slide 151 / 168
Slide 152 / 168
64
Given the measure of one angle, find the
measures of as many angles as possible.
What are the measures of angles 2, 4 and 8
respectively?
63
B 35 o , 35 o , 35
o
C 145 o , 35 o , 145
1
4
l
o
3
1
5
2
2
a
3
4
B Translation Only
n
8
7
8
A Reflection Only
7
b
6
5
m
Pull
0
t
Pull
A 55 o , 35 o , 55
If lines a and b are parallel, which transformation
justifies why
?
C Reflection and Translation
D The Angles are NOT Congruent
Slide 153 / 168
If lines a and b are parallel, which transformation
justifies why
?
66
If lines a and b are parallel, which transformation
justifies why
?
t
t
4
2
5
A Reflection Only
1
a
3
2
5
A Reflection Only
7
D The Angles are NOT Congruent
b
6
8
7
B Translation Only
B Translation Only
C Reflection and Translation
a
3
b
6
8
4
Pull
1
Pull
65
Slide 154 / 168
C Reflection and Translation
D The Angles are NOT Congruent
Slide 155 / 168
Slide 156 / 168
We can use what we've learned to establish some
interesting information about triangles.
For example, the sum of the angles of a triangle =
180.
Let's see why!
Applying what we've learned to prove
some interesting math facts...
Given
B
A
C
Slide 157 / 168
Slide 158 / 168
Let's draw a line through B parallel to AC.
We then have a two parallel lines cut by a transversal.
Number the angles and use what you know to prove the sum of
the measures of the angles equals 180.
l
1
B
2
C
A
l
1
B
m
p
n
1.
since if 2 parallel lines are cut by a transversal,
then alternate interior angles are congruent.
2
A
C
m
p
n
2.
is supplementary with
since if 2 parallel lines are
cut by a transversal, then same side interior angles are
supplementary.
3. Therefore,
Slide 159 / 168
Slide 160 / 168
Let's look at this another way...
2
B
Let's prove the Exterior Angle Theorem The measure of the exterior angle of a triangle is
equal to the sum of the remote interior angles.
l
1
C
A
B
m
p
n
1
1.
and
since if 2 parallel lines are cut by a
transversal, then alternate interior angles are congruent.
Remote Interior Angles
Slide 161 / 168
Slide 162 / 168
Let's draw a line through B parallel to AC.
We then have a two parallel lines cut by a transversal.
Number the angles and use what you know to prove the measure
of angle 1 = the sum of the measures of angles B and C.
B
l
2
3
1
l
2
C
Exterior Angle
2. Since all three angles form a straight line, the sum of the
angles is
B
A
n
A
C
m
p
1.
since if 2 parallel lines are cut by a transversal,
then alternate interior angles are congruent.
1
n
A
C
m
p
2.
3.
since if 2 parallel lines are cut by a transversal,
then alternate interior angles are congruent.
4. Therefore,
Slide 163 / 168
Slide 164 / 168
Example
Example
What is the measure of angle v in the diagram below?
What angles are congruent to angle 9?
1
6
2
3
p
5 4
v
10
7
9
11 12
14 13
8
r
h
g
Click
Slide 165 / 168
Slide 166 / 168
Example
67 What
is the measure of angle q in the diagram below?
Name the pairs of angles whose sum is congruent to angle
9.
10
7
9
3
p
5 4
q
11 12
14 13
8
r
Pull
2
1
6
h
g
Click
and
Slide 167 / 168
Slide 168 / 168
the expression that will make the statement
below true:
What is the measure of angle
7?
2
10
g
7
9
8
2
3
p
5 4
11 12
14 13
h
A
C
B
D
6
Pull
1
6
10
r
g
7
9
8
p
5 4
11 12
13
h
r
Pull
69
68 Choose