Transformations Rotations Reflections Dilations Symmetry Return - - PDF document

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8th Grade Math 2D Geometry: Transformations

www.njctl.org 2013-12-09

Slide 3 / 168 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

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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. A B C A' B' C' pre-image image Pull Pull

f

• r

t r a n s f

• r

m a t i

• n

s h

• w

n

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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 A B C X Y 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.

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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 C D 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? PULL

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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

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Translate 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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A B 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 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) 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) 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)

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Translating left/right changes 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) 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) 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)

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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 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) 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) 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) (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)

D E F G D' E' F' G' P u l l P u l l

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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 E F G D' E' F' G' P u l l P u l l

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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)

D E F G D' E' F' G' P u l l P u l l

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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 E F G D' E' F' G' P u l l P u l l

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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 E F G D' E' F' G' P u l l P u l l

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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

Slide 31 / 168 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 C D A' B' C' D' How is this figure rotated about the

• rigin?

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

• f rotation to

pre-image point, the

• ther 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 B B' D' E' D E P u l l P u l l 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 C D A' B' C' D' P u l l P u l l Check to see if the pre-image and image are congruent.

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When rotated counter-clockwise, the x-coordinate is the

• pposite 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 C D 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? Pull Pull

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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? Pull Pull A B C D 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) Pull Pull

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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) Pull Pull

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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) Pull Pull

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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) Pull Pull

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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) Pull Pull

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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'. Check to see if the pre-image and image are congruent.

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x y

A B C D

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 C D

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 C D

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

A B C D

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 E F

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. A B 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 C D Check to see if the pre-image and image are congruent.

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13 The reflection below represents a reflection across:

A the x axis B the y axis C the x axis, then the y axis D the y axis, then the x axis

x y

A B C A' B' C'

P u l l P u l l Check to see if the pre-image and image are congruent.

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14 The reflection below represents a reflection across:

A the x axis B the y axis C the x axis, then the y axis D the y axis, then the y axis

x y

D A B C A'

C' B' D'

P u l l P u l l Check to see if the pre-image and image are congruent.

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15 Which of the following represents a single reflection

• f Figure 1?

A B

C

D

Figure 1

P u l l P u l l

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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 P u l l P u l l

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17 Describe the reflection below:

A across the line y = x B across the y axis C across the line y = -3 D across the x axis

x y

A B C A' C' B' D'

P u l l P u l l

E' D E

Check to see if the pre-image and image are congruent.

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18 Describe the reflection below:

A across the line y = x B across the x axis C across the line y = -3 D across the line x = 4

x y

A B C A' C' B'

P u l l P u l l Check to see if the pre-image and image are congruent.

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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.

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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 image base 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). A A' 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 y

A 2 B 3 C

• 3

D 4 Pull Pull

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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) Pull Pull

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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) Pull Pull

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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) Pull Pull

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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 3 B

• 3

C 1/3 D

• 1/3

Pull Pull

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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 4

B

• 4

C 1/4 D

• 1/4

Pull Pull

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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 Pull Pull

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26 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 Pull Pull

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27Which of the following figures represents a reflection?

A Figure A B Figure B C Figure C D Figure D Pull Pull

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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 Pull Pull

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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 Pull Pull

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Symmetry

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Slide 81 / 168 Slide 82 / 168 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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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 Pull Pull

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31Which figure's dotted line shows a line of symmetry?

A B C D

Pull Pull

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32 Which of the objects does not have rotational symmetry?

A B C D Pull Pull 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 Similarity

Congruent shapes have the same size and shape. 2 figures are congruent if the second figure can be

• btained 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. Pull Pull

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Click for web page

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j

Example

What 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 Pull Pull

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34Which pair of shapes is similar but not congruent?

A B C D Pull Pull

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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 Pull Pull

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36Which of the following terms best describes the pair of

figures? A congruent B similar C neither congruent nor similar Pull Pull

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37 Which of the following terms best describes the 


pair of figures? A congruent B similar C neither congruent nor similar Pull Pull

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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

• ther through a series of translations, rotations, reflections

and/or dilations. The pre-image is dotted, the image is solid. Pull Pull

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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

• ther through a series of translations, rotations, reflections

and/or dilations. The pre-image is dotted, the image is solid. Pull Pull

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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

• ther through a series of translations, rotations, reflections

and/or dilations. The pre-image is dotted, the image is solid. Pull Pull

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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

• ther through a series of translations, rotations, reflections

and/or dilations. The pre-image is dotted, the image is solid. Pull Pull

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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.

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Vertical Angles are two angles that are opposite each

• ther when two lines intersect.

a b c d In this example, the vertical angles are: Vertical angles have the same measurement. So:

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x c a b d Vertical Angles can further be explained using the transformation

• f 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 b c a 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 Slide 112 / 168

Using what you know about complementary, supplementary and vertical angles, find the measure of the missing angles. b c a

By Vertical Angles: By Supplementary Angles:

Click Click

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38Are angles 2 and 4 vertical angles?

Yes No 1 2 3 4 Pull Pull

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39 Are angles 2 and 3 vertical angles?

Yes No 1 2 3 4 Pull Pull

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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.

2 1 3 4 Pull Pull

A 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.

2 1 3 4 Pull Pull

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

• ther 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! a b a b a b Adjacent Not Adjacent Not Adjacent

click 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 2

3 4

5 6

Pull Pull

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43 Which two angles are adjacent to each other?

A 3 and 6 B 5 and 4

1 2

3 4

5 6

Pull Pull

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Interactive Activity-Click Here

A P Q R B A E F

A transversal is a line that cuts across two or more (usually parallel) lines.

Slide 122 / 168 Recall From 3rd Grade

Shapes 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.

a b c d e f g h

Transversal

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 8

1 2 3 4 5 6 7 8

Pull Pull

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45 Which are pairs of corresponding angles?

A 2 and 6 B 3 and 1 C 1 and 8

1 2 3 4 5 6 7 8

Pull Pull

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46 Which are pairs of corresponding angles?

A 1 and 5 B 2 and 8 C 4 and 8

1 2 3 4 5 6 7 8

Pull Pull

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47 Which are pairs of corresponding


 angles ?

1 2 3 4 5 6 7 8

Pull Pull 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.

a b c d e f g h

In this diagram the alternate interior angles are: m n l

Click

Slide 129 / 168

Same Side Interior Angles are on same sides of the transversal and on the inside of the given lines.

a b 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.

a b 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 No 1 3 5 7 2 4 6 8 m n l Pull Pull

Slide 132 / 168

49 Are angles 3 and 6 alternate exterior angles?

Yes No Pull Pull 1 3 5 7 2 4 6 8 m n l

Slide 133 / 168

50 Are angles 7 and 4 alternate exterior angles?

Yes No Pull Pull 1 3 5 7 2 4 6 8 m n l

Slide 134 / 168

51 Which angle corresponds to angle 5?

A B C D 1

3 5 7 2 4 6 8 m n l Pull Pull

Slide 135 / 168

52Which pair of angles are same side interior?

A B C D 1

3 5 7 2 4 6 8 m l n Pull Pull

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 3 5 7 2 4 6 m n l 8

E Same Side Interior Pull Pull

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 3 5 7 2 4 6 m n l 8

E Same Side Interior Pull Pull

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 3 5 7 2 4 6 8 m n l

E Same Side Interior Pull Pull

Slide 139 / 168

56 Are angles 5 and 2 alternate interior angles?

Yes No Pull Pull 1 3 5 7 2 4 6 8 m n l

Slide 140 / 168

57 Are angles 5 and 7 alternate interior angles?

Yes No Pull Pull 5 7 2 4 6 8 n 1 3 m l

Slide 141 / 168

58 Are angles 7 and 2 alternate interior angles?

1 3 5 7 2 4 6 8 m n l Yes No Pull Pull

Slide 142 / 168

59 Are angles 3 and 6 alternate exterior angles?

Yes No Pull Pull 1 3 5 7 2 4 8 m n l 6

Slide 143 / 168

1 3 5 7 2 4 6 8

l m n are supplementary are supplementary

These Special Cases can further be explained using the transformations of reflections and translations

Special Cases

If 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 3 5 7 2 4 6 8

l m n a b

Line a cuts angles 3 and 5 in half. When angle 1 is reflected

• ver line a, it forms angle 7.

When angle 7 is reflected

• ver line a, it forms angle 1.

Line b cuts angles 4 and 6 in half. When angle 2 is reflected

• ver line b, it forms angle 8.

When angle 8 is reflected

• ver line b, it forms angle 2.

Reflections

Slide 145 / 168

1 3 5 7 2 4 6 8

l m n d c

Reflections Continued

Line d cuts angles 2 and 8 in half. When angle 4 is reflected

• ver line d, it forms angle 6.

When angle 6 is reflected

• ver line d, it forms angle 4.

Line c cuts angles 1 and 7 in half. When angle 3 is reflected

• ver line c, it forms angle 5.

When angle 5 is reflected

• ver line c, it forms angle 3.

Slide 146 / 168 Translations

1 3 5 7

m

2 4 6 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 4 6 8

l n

1 3 5 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 4 6 8

l n

1 3 5 7

m

The translations also work if line l is translated y units up and x units right.

1 3 5 7

m

2 4 6 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 5 6 2 7 1 8

l m n Pull Pull A

<4, <5, <6 B <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? Pull Pull

4 5 6 2 7 1 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? Pull Pull

1 3 5 7 2 4 8 m n l

A <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 4 8 m n l

Pull Pull A 55 o, 35 o, 55 0 B 35 o, 35 o, 35 o C 145 o, 35 o, 145

• Slide 152 / 168

64 If lines a and b are parallel, which transformation

justifies why ?

A Reflection Only B Translation Only C Reflection and Translation D The Angles are NOT Congruent

1 3 5 7 2 4 6 8

b a t Pull Pull

Slide 153 / 168

65 If lines a and b are parallel, which transformation

justifies why ?

A Reflection Only B Translation Only C Reflection and Translation D The Angles are NOT Congruent

1 3 5 7 2 4 6 8

b a t Pull Pull

Slide 154 / 168

66 If lines a and b are parallel, which transformation

justifies why ?

A Reflection Only B Translation Only C Reflection and Translation D The Angles are NOT Congruent Pull Pull

1 3 5 7 2 4 6 8

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 1

2

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 C 1 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

• f angle 1 = the sum of the measures of angles B and C.

l m n p B A C 1 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 C 1 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 3 4 5 6 7 8 9 10 11 12 13 14 What angles are congruent to angle 9? Click

Slide 165 / 168 Example

p r g h 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Name the pairs of angles whose sum is congruent to angle 9. and Click

Slide 166 / 168

67What is the measure of angle q in the diagram below? q

Pull Pull

Slide 167 / 168

68Choose the expression that will make the statement

below true:

A B C D

p r g h 1 2 3 4 5 6 7 8 9 10 11 12 13 14

Pull Pull

Slide 168 / 168

69 What is the measure of angle

7?

p r g h 2 4 5 6 7 8 9 10 11 12 13 Pull Pull