Transformations 8th Grade Math 2D Geometry: Transformations · 8th Grade Math 2D Geometry: Transformations 2013-12-09 Slide 3 / 168 Table of Contents · Reflections ... rule = (x

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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


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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.

page1svg

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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


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


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?


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)

http://www.mathwarehouse.com/transformations/translations-interactive-activity.php

http://www.mathwarehouse.com/transformations/translations-interactive-activity.php

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Translating left/right changes the x-coordinate.

Translating up/down changes the y-coordinate.





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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.





(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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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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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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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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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


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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


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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


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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


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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'.


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x

y

A B

CD

Reflect the figure across the y-axis.


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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.


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.


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.


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x

y

A B

C D

EF

Reflect the figure across the y-axis.Click to see reflection.


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x

y

Reflect the figure across the line x = -2.

AB

C

D

E


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x

y

Reflect the figure across the line y = x.

A B

CD


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


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


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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


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


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Dilations


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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.

page1svg

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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)


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)


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


PullPull

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28 Which of the following figures represents a dilation?A Figure A B Figure B


PullPull

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29 Which of the following figures represents a translation?A Figure A B Figure B


PullPull

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Symmetry


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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?

page1svg

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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

http://www.symmeter.com/symfacer.htm

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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


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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

page1svg

http://www.nctm.org/standards/content.aspx?id=26770

http://www.nctm.org/standards/content.aspx?id=26770

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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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PullPull

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PullPull

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PullPull

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Special Pairs of Angles


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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.

page1svg

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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.

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Recall From 3rd GradeShapes and Perimeters

Parallel lines are a set of two lines that do not intersect (touch).

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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

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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

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A 2 and 6

B 3 and 1

C 1 and 8

1

23

4

56

78

PullPull

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A 1 and 5

B 2 and 8

C 4 and 8

1 2

3 4

56

7 8

PullPull

http://www.mathopenref.com/transversal.html

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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

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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

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48 Are angles 2 and 7 alternate exterior angles?

Yes

No1 3

5 7

2 46 8

m

n

lPullPull

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Yes

No PullPull

1 3

5 7

2 46 8

m

n

l

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Yes

No PullPull

1 3

5 7

2 46 8

m

n

l

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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

Slide 136 / 168

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


A Alternate Interior Angles B Alternate Exterior Angles C Corresponding Angles D Vertical Angles

1 35 7

2 46

m

n

l

8


PullPull

Slide 138 / 168


A Alternate Interior Angles B Alternate Exterior Angles C Corresponding Angles D Vertical Angles

1 35 7

2 46 8

m

n

l


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


Yes

No PullPull

5 7

2 46 8

n

1 3 m

l

Slide 141 / 168


1 3

5 7

2 46 8

m

n

lYes

NoPullPull

Slide 142 / 168


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

Slide 145 / 168

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.

Slide 146 / 168

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

Slide 147 / 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.

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

Slide 148 / 168

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

Slide 149 / 168

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

Slide 150 / 168

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

Slide 152 / 168

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 ?


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 ?


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

Slide 157 / 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

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


2.


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

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Transformations 8th Grade Math 2D Geometry: Transformations · 8th Grade Math 2D Geometry: Transformations 2013-12-09 Slide 3 / 168 Table of Contents · Reflections ... rule = (x