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New Jersey Center for Teaching and Learning
Progressive Mathematics Initiative
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8th Grade Math
2D Geometry: Transformations
www.njctl.org
2013-12-09
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Table of Contents
· Reflections· Dilations
· Translations
Click on a topic to go to that section
· Rotations
· Transformations
· Congruence & Similarity
Common Core Standards: 8.G.1, 8.G.2, 8.G.3, 8.G.4, 8.G.5
· Special Pairs of Angles
· Symmetry
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Transformations
Return to Table of Contents
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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.
AB
C
A'B'
C'
pre-image image
Pull
Pull
for transformation shown
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The image can also be labeled with new letters as shown below.
Triangle ABC is the pre-image to the reflected image triangle XYZ
AB
C
XY
Z
pre-image image
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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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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.
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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Click for web page
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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.
A B
CD
A' B'
C'D'
To complete a translation, move each point of the pre-image and label the new point.
Example: Move the figure left 2 units and up 5 units. What are the coordinates of the pre-image and image? PU
LL
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Translate pre-image ABC 2 left and 6 down. What are the coordinates of the image and pre-image?
A
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.
PULL
Slide 16 / 168Translate pre-image ABCD 4 right and 1 down. What are the coordinates of the image and pre-image?
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.
PULL
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AB
C
D
Translate pre-image ABCD 5 left and 3 up.
What are the coordinates of the image and pre-image?
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.
PULL
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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.
2 Left and 5 UpA (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 DownA (-2,7) A' (-4,1)B (-3,1) B' (-5,-5)C (-6,3) C' (-8,-3)
4 Right and 1 DownA (-5,4) A' (-1,3)B (-1,2) B' (3,1)C (-4,-2) C' (0,-3)D (-6, 1) D' (-2,0)
5 Left and 3 UpA (3,2) A' (-2,5)B (7,1) B' (2,4)C (4,0) C' (-1,3)D (2,-2) D' (-3,1)
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Translating left/right changes the x-coordinate.
Translating up/down changes the y-coordinate.
2 Left and 5 UpA (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 DownA (-2,7) A' (-4,1)B (-3,1) B' (-5,-5)C (-6,3) C' (-8,-3)
4 Right and 1 DownA (-5,4) A' (-1,3)B (-1,2) B' (3,1)C (-4,-2) C' (0,-3)D (-6, 1) D' (-2,0)
5 Left and 3 UpA (3,2) A' (-2,5)B (7,1) B' (2,4)C (4,0) C' (-1,3)D (2,-2) D' (-3,1)
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Translating left/right changes the x-coordinate.· Left subtracts from the x-coordinate
· 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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2 units Left … x-coordinate - 2 y-coordinate stays rule = (x - 2, y)
5 units Right & 3 units Down… x-coordinate + 5 y-coordinate - 3 rule = (x + 5, y - 3)
click to reveal
A rule can be written to describe translations on the coordinate plane.
click to reveal
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Write a rule for each translation.
2 Left and 5 UpA (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 DownA (-2,7) A' (-4,1)B (-3,1) B' (-5,-5)C (-6,3) C' (-8,-3)
4 Right and 1 DownA (-5,4) A' (-1,3)B (-1,2) B' (3,1)C (-4,-2) C' (0,-3)D (-6, 1) D' (-2,0)
5 Left and 3 UpA (3,2) A' (-2,5)B (7,1) B' (2,4)C (4,0) C' (-1,3)D (2,-2) D' (-3,1)
(x, y) (x-2, y+5) (x, y) (x-2, y-6)
(x, y) (x-5, y+3) (x, y) (x+4, y-1)
click to reveal click to reveal
click to reveal click to reveal
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1 What rule describes the translation shown?
A (x,y) (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)
DE
F
G
D'E'
F'
G' PullPull
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2 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
EF
G
D'E'
F'
G'
PullPull
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3 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)
DE
F
G
D'E'
F'
G'
PullPull
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4 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
EF
G
D' E'F'
G'
PullPull
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5 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
EF
G
D'E'
F'
G'
PullPull
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Rotations
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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Rotation
The person's finger is the point of rotation for each figure.
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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.
A
This figure is rotated 90 degrees
counterclockwise about point A.
B
This figure is rotated 180 degrees
clockwise about point B.
Click for answer Click for answer
Hint
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A B
CD
A'
B' C'
D'
How is this figure rotated about the origin?
In a coordinate plane, each quadrant represents
This figure is rotated 270 degrees clockwise about the origin or 90 degrees counterclockwise about the origin.
Click to Reveal
Check to see if the pre-image and image are congruent.
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.
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The following descriptions describe the same rotation. What do you notice? Can you give your own example?
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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.)
A clockwise
B counterclockwise
C 90 degrees
D 180 degrees
E 270 degrees
A, A'C'
C
BB'
D'E'
D
EPullPull
Check to see if the pre-image and image are congruent.
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7 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
CD
A'B'
C' D'
PullPull
Check to see if the pre-image and image are congruent.
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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:
(x, y) (-y, x)
Click to Reveal
A B
CD
A'
B' C'
D'
Now let's look at the same figure and see what happens to the coordinates when we rotate a figure.
Write the coordinates for the pre-image and image.
What do you notice?
PullPull
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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. In other words:
(x, y) (-x, -y)
Click to Reveal
What happens to the coordinates in a half-turn?
Write the coordinates for the pre-image and image.
What do you notice?
PullPullA B
CD
A'B'
C' D'
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Can you summarize what happens to the coordinates during a rotation?
Counterclockwise:
Half-turn:
Clockwise:
(x, y) (-y, x)
(x, y) (y, -x)
(x, y) (-x, -y)
Click to Reveal
Click to Reveal
Click to Reveal
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8 What are the new coordinates of a point A (5, -6) after a rotation clockwise?
A (-6, -5)
B (6, -5)
C (-5, 6)
D (5, -6)
PullPull
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9 What are the new coordinates of a point S (-8, -1) after a rotation counterclockwise?
A (-1, -8)
B (1, -8)
C (-1, 8)
D (8, 1)
PullPull
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10 What are the new coordinates of a point H (-5, 4) after a rotation counterclockwise?
A (-5, -4)
B (5, -4)
C (4, -5)
D (-4, 5)
PullPull
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11 What are the new coordinates of a point R (-4, -2) after a rotation clockwise?
A (2, -4)
B (-2, 4)
C (2, 4)
D (-4, 2)
PullPull
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12 What are the new coordinates of a point Y (9, -12) after a half-turn?
A (-12, 9)
B (-9,12)
C (-12, -9)
D (9,12)
PullPull
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Reflections
Return to Table of Contents
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Examples
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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.
A
B C
A'
B'C'
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.
Each vertex in ABC is the same distance from line t as the vertices in A'B'C'.
Check to see if the pre-image and image are congruent.
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x
y
A B
CD
Reflect the figure across the y-axis.
Check to see if the pre-image and image are congruent.
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x
y
A B
CD
What do you notice about the coordinates when you reflect across the y-axis?
A'B'
C'D'
A (-6, 5) A' (6, 5)B (-4, 5) B' (4, 5)C (-4, 1) C' (4, 1)D (-6, 3) 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 y-axis.
Check to see if the pre-image and image are congruent.
Tap box for coordinates
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x
y
A B
CD
What do you predict about the coordinates when you reflect across the x-axis?
A' B'
C'D'
A (-6, 5) A' (-6, -5)B (-4, 5) B' (-4, -5)C (-4, 1) C' (-4, -1)D (-6, 3) D' (-6, -3)
When you reflect across the x-axis, the y-coordinate becomes the opposite.
So (x, y) (x, -y) when you reflect across the x-axis.
Check to see if the pre-image and image are congruent.
Tap box for coordinates
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x
y
AB
CD
Reflect the figure across the y-axis then the x-axis.Click to see each reflection.
Check to see if the pre-image and image are congruent.
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x
y
A B
C D
EF
Reflect the figure across the y-axis.Click to see reflection.
Check to see if the pre-image and image are congruent.
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x
y
Reflect the figure across the line x = -2.
AB
C
D
E
Check to see if the pre-image and image are congruent.
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x
y
Reflect the figure across the line y = x.
A B
CD
Check to see if the pre-image and image are congruent.
Slide 57 / 16813 The reflection below represents a reflection across:
A the x axis
B the y axisC the x axis, then the y axis
D the y axis, then the x axis
x
y
A
B C
A'
B' C'
PullPull
Check to see if the pre-image and image are congruent.
Slide 58 / 16814 The reflection below represents a reflection across:
A the x axis
B the y axisC the x axis, then the y axis
D the y axis, then the y axis
x
yDA
B C
A'
C' B'
D'
PullPull
Check to see if the pre-image and image are congruent.
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15 Which of the following represents a single reflection of Figure 1?
A
B
C
D
Figure 1
PullPull
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16 Which of the following describes the movement below?
A reflection
B rotation, 90 clockwise
C slide
D rotation, 180 clockwise
PullPull
Slide 61 / 16817 Describe the reflection below:
A across the line y = x
B across the y axisC across the line y = -3
D across the x axis
x
y
A
B CA'
C'
B'
D'PullPull
E'
DE
Check to see if the pre-image and image are congruent.
Slide 62 / 16818 Describe the reflection below:
A across the line y = x
B across the x axisC across the line y = -3D across the line x = 4
x
y
A
B
C
A'
C'
B'
PullPull
Check to see if the pre-image and image are congruent.
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Dilations
Return to Table of Contents
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A dilation is 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.
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The scale factor is the ratio of sides:
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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x
y
Example.
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?
This is an enlargement.Scale Factor is 2.
base length of imagebase length of pre-image
6 3 = 2
Click to reveal
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x
y
What happened to the coordinates with a scale factor of 2?
A (0, 1) A' (0, 2)B (3, 2) B' (6, 4)C (4, 0) C' (8, 0)D (1, 0) D' (2, 0)
The coordinates were all multiplied by 2.
The center for this dilation was the origin (0,0).
AA' B
B'
C C'DD'Click to reveal
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19What is the scale factor for the image shown below? The pre-image is dotted and the image is solid.
x
yA 2
B 3
C -3
D 4
PullPull
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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?
A (12, -8)
B (-12, -8)
C (-12, 8)
D (-3/4, 1/2)
PullPull
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21What are the coordinates of a point Y (-2, 5) after a dilation with a scale factor of 2.5?
A (-0.8, 2)
B (-5, 12.5)
C (0.8, -2)
D (5, -12.5)
PullPull
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22What are the coordinates of a point X (4, -8) after a dilation with a scale factor of 0.5?
A (-8, 16)
B (8, -16)
C (-2, 4)
D (2, -4)
PullPull
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23 The coordinates of a point change as follows during a dilation: (-6, 3) (-2, 1)
What is the scale factor?
A 3B -3C 1/3
D -1/3
PullPull
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24The coordinates of a point change as follows during a dilation:
(4, -9) (16, -36)
What is the scale factor?
A 4B -4
C 1/4
D -1/4
PullPull
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25The coordinates of a point change as follows during a dilation:
(5, -2) (17.5, -7)
What is the scale factor?
A 3
B -3.75
C -3.5
D 3.5
PullPull
Slide 76 / 16826 Which of the following figures represents a rotation?
(and could not have been achieved only using a reflection)A Figure A B Figure B
C Figure C D Figure D
PullPull
Slide 77 / 16827Which of the following figures represents a reflection?
A Figure A B Figure B
C Figure C D Figure D
PullPull
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28 Which of the following figures represents a dilation?A Figure A B Figure B
C Figure C D Figure D
PullPull
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29 Which of the following figures represents a translation?A Figure A B Figure B
C Figure C D Figure D
PullPull
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Symmetry
Return to Table of Contents
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SymmetryA 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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Will Smith with a symmetrical face.
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.
Marilyn Monroe with a
symmetrical face.
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Click the picture below to learn how to make your own face symmetrical.
Tina Fey
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Rotational symmetry is when a figure can be rotated around a point onto itself in less than a 360 turn.
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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?
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30 How many lines of symmetry does this figure have?
A 3
B 6
C 5
D 4
PullPull
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31Which figure's dotted line shows a line of symmetry?
A B C D PullPull
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32 Which of the objects does not have rotational symmetry?
A
B
C
D
PullPull
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 &Similarity
Return to Table of Contents
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Congruence and SimilarityCongruent 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.
Remember - translations, reflections and rotations preserve image size and shape.
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Similar shapes have the same shape, congruent angles and proportional sides.
2 figures are similar if the second figure can be obtained from the first by a series of translations, reflections, rotations and/or dilations.
PullPull
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Click for web page
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j
ExampleWhat would the measure of angle j have to be in order for the figures below to be similar?
180 - 112 - 33 = 35
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Are the two triangles below similar? Explain your reasoning?
Example
Yes, the triangles have congruent angles and are therefore similar.Click
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33Which pair of shapes is similar but not congruent?
A
B
C
D
PullPull
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34Which pair of shapes is similar but not congruent?
A
B
C
D
PullPull
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35 Which of the following terms best describes the pair of figures?
A congruent
B similar
C neither congruent nor similar
PullPull
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36Which of the following terms best describes the pair of figures?
A congruent
B similar
C neither congruent nor similar PullPull
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37 Which of the following terms best describes the pair of figures?
A congruent
B similar
C neither congruent nor similarPullPull
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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.
PullPull
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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.
PullPull
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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.
PullPull
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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.
PullPull
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Special Pairs of Angles
Return to Table of Contents
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Recall:
· Complementary Angles are two angles with a sum of 90 degrees.
These two angles are complementary angles because their sum is 90.
Notice that they form a right angle when placed together.
· Supplementary Angles are two angles with a sum of 180 degrees.
These two angles are supplementary angles because their sum is 180.
Notice that they form a straight angle when placed together.
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Vertical Angles are two angles that are opposite each other when two lines intersect.
a bcd
In this example, the vertical angles are:
Vertical angles have the same measurement. So:
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xc
ab d
Vertical Angles can further be explained using the transformation of reflection.
Transformations
Line x cuts angles b and d in half.
When angle a is reflected over line x, it forms angle c.
When angle c is reflected over line x, it forms angle a.
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y
bca
d
Line y cuts angles a and c in half.
When angle b is reflected over line y, it forms angle d.
When angle d is reflected over line y, it forms angle b.
Transformations Continued
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Using what you know about complementary, supplementary and vertical angles, find the measure of the missing angles.
bc
a
By Vertical Angles: By Supplementary Angles:
Click Click
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38Are angles 2 and 4 vertical angles?
Yes
No
12
34
PullPull
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39 Are angles 2 and 3 vertical angles?
Yes
No
12
34
PullPull
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40 If angle 1 is 60 degrees, what is the measure of angle 3? You must be able to explain why.
21 3
4
PullPullA 30 o
B 60 o
C 120 o
D 15 o
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41 If angle 1 is 60 degrees, what is the measure of angle 2? You must be able to explain why.
21
34
PullPull
A 30 o
B 60 o
C 120 o
D 15 o
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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.
A
B
C
D
is adjacent to
How do you know?· They have a common side (ray )· They have a common vertex (point B)
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Adjacent or Not Adjacent? You Decide!
ab a
b
a
b
Adjacent Not Adjacent Not Adjacentclick to reveal click to reveal click to reveal
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42 Which two angles are adjacent to each other?
A 1 and 4
B 2 and 4
1
23
456
PullPull
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43 Which two angles are adjacent to each other?
A 3 and 6
B 5 and 4
12
34 5
6
PullPull
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Interactive Activity-Click Here
A
PQ
RB
A
E
F
A transversal is a line that cuts across two or more (usually parallel) lines.
Slide 122 / 168
Recall From 3rd GradeShapes and Perimeters
Parallel lines are a set of two lines that do not intersect (touch).
Slide 123 / 168
Corresponding Angles are on the same side of the transversal and on the same side of the given lines.
ab
c d
e f
g h
Tran
sver
sal
In this diagram the corresponding angles are:
Click
Slide 124 / 168
44 Which are pairs of corresponding angles?
A 2 and 6
B 3 and 7
C 1 and 81 2
3 4
5 6
7 8
PullPull
Slide 125 / 168
45 Which are pairs of corresponding angles?
A 2 and 6
B 3 and 1
C 1 and 8
1
23
4
56
78
PullPull
Slide 126 / 168
46 Which are pairs of corresponding angles?
A 1 and 5
B 2 and 8
C 4 and 8
1 2
3 4
56
7 8
PullPull
Slide 127 / 168
47 Which are pairs of corresponding angles ?
1
2
3
45
6
7
8 PullPull
A 2 and 4
B 6 and 5
C 7 and 8
D 1 and 3
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Alternate Interior Angles are on opposite sides of the transversal and on the inside of the given lines.
ab
c d
e f
g h
In this diagram the alternate interior angles are: m
n
l
Click
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Same Side Interior Angles are on same sides of the transversal and on the inside of the given lines.
ab
c d
e f
g h
m
n
l
In this diagram the same side interior angles are:
Click
Slide 130 / 168
Alternate Exterior Angles are on opposite sides of the transversal and on the outside of the given lines.
ab
c d
e f
g h
In this diagram the alternate exterior angles are:
l
m
n
Which line is the transversal?
Click
Slide 131 / 168
48 Are angles 2 and 7 alternate exterior angles?
Yes
No1 3
5 7
2 46 8
m
n
lPullPull
Slide 132 / 168
49 Are angles 3 and 6 alternate exterior angles?
Yes
No PullPull
1 3
5 7
2 46 8
m
n
l
Slide 133 / 168
50 Are angles 7 and 4 alternate exterior angles?
Yes
No PullPull
1 3
5 7
2 46 8
m
n
l
Slide 134 / 168
51 Which angle corresponds to angle 5?
AB
C
D1 3
5 7
2 46 8
m
n
l
PullPull
Slide 135 / 168
52Which pair of angles are same side interior?
AB
C
D1 3
5 7
2 46 8
m
l
n
PullPull
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53 What type of angles are and ?
A Alternate Interior Angles
B Alternate Exterior Angles C Corresponding Angles D Vertical Angles
1 35 7
2 4
6
m
n
l
8
E Same Side Interior
PullPull
Slide 137 / 168
54 What type of angles are and ?
A Alternate Interior Angles B Alternate Exterior Angles C Corresponding Angles D Vertical Angles
1 35 7
2 46
m
n
l
8
E Same Side Interior
PullPull
Slide 138 / 168
55 What type of angles are and ?
A Alternate Interior Angles B Alternate Exterior Angles C Corresponding Angles D Vertical Angles
1 35 7
2 46 8
m
n
l
E Same Side Interior
PullPull
Slide 139 / 168
56 Are angles 5 and 2 alternate interior angles?
Yes
No PullPull
1 35 7
2 46 8
m
n
l
Slide 140 / 168
57 Are angles 5 and 7 alternate interior angles?
Yes
No PullPull
5 7
2 46 8
n
1 3 m
l
Slide 141 / 168
58 Are angles 7 and 2 alternate interior angles?
1 3
5 7
2 46 8
m
n
lYes
NoPullPull
Slide 142 / 168
59 Are angles 3 and 6 alternate exterior angles?
Yes
NoPullPull
1 3
5 7
2 48
m
n
l
6
Slide 143 / 168
1 35 7
2 46 8
l
m
n
are supplementary
are supplementary
These Special Cases can further be explained using the transformations of reflections and translations
Special CasesIf parallel lines are cut by a transversal then:
· Corresponding Angles are congruent
· Alternate Interior Angles are congruent
· Alternate Exterior Angles are congruent
· Same Side Interior Angles are supplementary
SO:
are supplementary
are supplementary
click
Slide 144 / 168
1 35 7
2 46 8
l
m
na
b
Line a cuts angles 3 and 5 in half.
When angle 1 is reflected over line a, it forms angle 7.
When angle 7 is reflected over line a, it forms angle 1.
Line b cuts angles 4 and 6 in half.
When angle 2 is reflected over line b, it forms angle 8.
When angle 8 is reflected over line b, it forms angle 2.
Reflections
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1 35 7
2 46 8
l
m
n
d
c
Reflections Continued
Line d cuts angles 2 and 8 in half.
When angle 4 is reflected over line d, it forms angle 6.
When angle 6 is reflected over line d, it forms angle 4.
Line c cuts angles 1 and 7 in half.
When angle 3 is reflected over line c, it forms angle 5.
When angle 5 is reflected over line c, it forms angle 3.
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Translations1 3
5 7
m
2 46 8
l
n
Line m is parallel to line l.
If line m is translated y units down, it will overlap with line l.
2 46 8
l
n
1 35 7
m
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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.
2 46 8
l
n
1 35 7
m
The translations also work if line l is translated y units up and x units right.
1 35 7
m2 46 8
l
n
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60 Given the measure of one angle, find the measures of as many angles as possible.Which angles are congruent to the given angle?
4 56
2 71 8
l
m
n
PullPull
A <4, <5, <6B <5, <7, <1
C <2
D <5, <1
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61 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?
Pul
lP
ull4 5
6
2 71 8
l
m
n
A 50 o
B 40 o
C 130 o
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62 Given the measure of one angle, find the measures of as many angles as possible.Which angles are congruent to the given angle?
PullPull1 3
5 7
2 48
m
n
lA <4
B <4, <5, <3
C <2
D <8
Slide 151 / 168
63 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?
1 3
5 7
2 48
m
n
l
Pul
lP
ull
A 55 o, 35 o, 55 0
B 35 o, 35 o, 35 o
C 145 o, 35 o, 145 o
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64 If lines a and b are parallel, which transformation justifies why ?
A Reflection OnlyB Translation OnlyC Reflection and TranslationD The Angles are NOT Congruent
13
57
24
68
b
a
t
Pul
lP
ull
Slide 153 / 16865 If lines a and b are parallel, which transformation
justifies why ?
A Reflection OnlyB Translation OnlyC Reflection and TranslationD The Angles are NOT Congruent
13
57
24
68
b
a
t
Pul
lP
ull
Slide 154 / 168
66 If lines a and b are parallel, which transformation justifies why ?
A Reflection OnlyB Translation OnlyC Reflection and TranslationD The Angles are NOT Congruent
Pul
lP
ull
13
57
24
68
b
a
t
Slide 155 / 168
Applying what we've learned to prove some interesting math facts...
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!
Given
B
A C
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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
m
n p
B
A C
1
2
Slide 158 / 168
l
m
n p
B
A C
1
2
1. since if 2 parallel lines are cut by a transversal, then alternate interior angles are congruent.
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
1. and since if 2 parallel lines are cut by a transversal, then alternate interior angles are congruent.
2. Since all three angles form a straight line, the sum of the angles is
l
m
n p
B
A C
12
Let's look at this another way...
Slide 160 / 168
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.
B
A C1
Exterior Angle
Remote Interior Angles
Slide 161 / 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.
l
m
n p
B
A C1
2
Slide 162 / 168
1. since if 2 parallel lines are cut by a transversal, then alternate interior angles are congruent.
2.
3. since if 2 parallel lines are cut by a transversal, then alternate interior angles are congruent.
4. Therefore,
l
m
n p
B
A C1
3
2
Slide 163 / 168
Example
v
What is the measure of angle v in the diagram below?
Slide 164 / 168
Example
p
r
g h
1 2 3456
7 8910
11 121314
What angles are congruent to angle 9?
Click
Slide 165 / 168
Example
p
r
g h
1 2 3456
7 8910
11 121314
Name the pairs of angles whose sum is congruent to angle 9.
andClick
Slide 166 / 168
67What is the measure of angle q in the diagram below?
q
Pul
lP
ull
Slide 167 / 168
68Choose the expression that will make the statement below true:
A
B
C
D
p
r
g h
1 2 3456
7 8910
11 121314
Pul
lP
ull
Slide 168 / 168
69 What is the measure of angle 7?
p
r
g h
2
456
7 8910
11 1213
Pul
lP
ull