Transformations Rotations Reflections Dilations Symmetry - - PDF document

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8th Grade

2D Geometry: Transformations

2015-11-13 www.njctl.org

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· Reflections · Dilations · Translations

Click on a topic to go to that section

· Rotations · Transformations · Congruence & Similarity · Special Pairs of Angles · Symmetry · Glossary · Remote Exterior Angles

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Transformations

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Any time you move, shrink, or enlarge a figure you make a transformation. A B C A' B' C' pre-image image

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 figure (image) with the same letters and the prime sign.

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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

Transformation

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There are four types of transformations in this unit: · Translations · Rotations · Reflections · Dilations

Transformations

The first three transformations preserve the size and shape of the

• figure. Therefore, both the pre-image and image will be congruent.

Congruent figures are same size and same shape. In other words: If your pre-image is a trapezoid, your image is a congruent trapezoid. If your pre-image is an angle, your image is an angle with the same measure. If your pre-image contains parallel lines, your image contains parallel lines.

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Translations

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Slide 9 / 230 Translations Slide 10 / 230

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.

Translation

You can use a slide arrow to show the direction and distance

• f 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.

Translation Slide 12 / 230

Click for web page

Translation

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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? click to reveal

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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.

click to reveal

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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. click to reveal

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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. Click

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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)

Translations Rule Slide 18 / 230

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)

Translations Rule

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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

Translations Rule Slide 20 / 230

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) A rule can be written to describe translations on the coordinate plane.

click click

Translations Rule Slide 21 / 230

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

Translations Rule

click to reveal click to reveal click to reveal click to reveal

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D E F G D' E' F' G'

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)

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D E F G D' E' F' G'

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)

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D E F G D' E' F' G'

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)

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D E F G D' E' F' G'

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)

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D E F G D' E' F' G'

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)

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Rotations

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Slide 28 / 230 Rotations Slide 29 / 230

A rotation (turn) moves a figure around a point. This point can be the index finger or it can be some other point. P

Rotations

This point is called the point of rotation.

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The person's finger is the point of rotation for each figure.

Rotations

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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. This figure is rotated 90º counterclockwise about point A. This figure is rotated 180º clockwise about point B.

click to reveal

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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 90º. 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?

Rotations Slide 34 / 230 Rotations

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

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

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

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Teachers: Use this Mathematical Practice Pull Tab for the next 3 example slides & the "General Formula" slide that follows.

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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 90º counterclockwise. Write the coordinates for the pre-image and image. What do you notice?

Rotations Slide 39 / 230

A B C D A' B' C' D' What happens to the coordinates in a half-turn? Write the coordinates for the pre-image and image. What do you notice?

Rotations Slide 40 / 230

A B C D A' B' C' D' What happens to the coordinates in a 90º clockwise? Write the coordinates for the pre-image and image. What do you notice?

Rotations Slide 41 / 230

Summarize what happens to the coordinates during a rotation? 90º Counterclockwise: Half-turn: 90º Clockwise:

Rotations Slide 42 / 230

8 What are the new coordinates of a point A (5, -6) after

90º rotation clockwise? A (-6, -5) B (6, -5) C (-5, 6) D (5, -6)

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9 What are the new coordinates of a point S (-8, -1) after a 90º rotation counterclockwise? A (-1, -8) B (1, -8) C (-1, 8) D (8, 1)

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10 What are the new coordinates of a point H (-5, 4) after a 180º rotation counterclockwise? A (-5, -4) B (5, -4) C (4, -5) D (-4, 5)

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11 What are the new coordinates of a point R (-4, -2) after a 270º rotation clockwise? A (2, -4) B (-2, 4) C (2, 4) D (-4, 2)

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12 What are the new coordinates of a point Y (9, -12) after a half-turn? A (-12, 9) B (-9,12) C (-12, -9) D (9,12)

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A B C D x y

13 Parallelogram A' B' C' D' (not shown)is the image of parallelogram ABCD after a rotation of 180º about the origin. Which statements about parallelogram A'B'C'D' are true? Select each correct statement. A A'B' is parallel to B'C' B A'B' is parallel to A'D' C A'B' is parallel to C'D' D A'D' is parallel to B'C' E A'D' is parallel to D'C'

From PARCC EOY sample test non-calculator #8

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14 Lines m and n are parallel on a coordinate plane. Lines m and n are transformed by the same rotation resulting in image lines s and

• t. Which statement describes the relationship between lines s and

t? A Lines s and t are parallel. B Lines s and t are perpendicular. C Lines s and t are intersecting but not perpendicular. D The relationship between lines s and t cannot be determined without knowing the angle of the rotation.

From PARCC PBA sample test non-calculator #7

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Reflections

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Slide 50 / 230 Reflections Slide 51 / 230

A reflection (flip) creates a mirror image of a figure.

Reflection Slide 52 / 230

A

B C A'

B'

C'

t

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 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 ΔABC is the same distance from line t as the vertices in ΔA'B'C'. Check to see if the pre-image and image are congruent.

Reflection Slide 53 / 230

x y

A B C D Reflect the figure across the y-axis. Check to see if the pre-image and image are congruent.

Reflection Slide 54 / 230

x y

A B C D A' B' C' D' What do you notice about the coordinates when you reflect across the y-axis?

Reflection

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x y

A B C D A' B' C' D' What do you predict about the coordinates when you reflect across the x-axis?

Reflection Slide 56 / 230

x y A B C D Reflect the figure across the y-axis then the x-axis. Click to see each reflection.

Example Slide 57 / 230

x y

A B C D E F Reflect the figure across the y-axis. Click to see reflection.

Example Slide 58 / 230

x y

A B C D E Reflect the figure across the line x = -2.

Example Slide 59 / 230

x y

A B C D Reflect the figure across the line y = x.

Example Slide 60 / 230

x y A B C A' B' C' 15 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

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x y D B C A A' C' B' D' 16 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

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17 Which of the following represents a single reflection of Figure 1? A B C D

Figure 1

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18 Which of the following describes the movement below? A reflection B rotation, 180º clockwise C slide D rotation, 90º clockwise

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x y A B C D E A' C' B' D' E' 19 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

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

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x y A B C A' C' B' 20 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

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

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Three congruent figures are shown on the coordinate

• plane. Use these figures to answer the next 2 response

questions.

From PARCC EOY sample test non-calculator #12

1 2 3 y x

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21 Part A

Select a transformation from each group of choices to make the statement true. Figure 1 can be transformed onto figure 2 by:

A a reflection across the x-axis B a rotation 180º clockwise about the origin C a translation 2 units to the left D a reflection across the y-axis E a rotation 90º clockwise about the origin F a translation 3 units to the right

followed by

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22 Part B

Figure 3 can also be created by transforming figure 1 with a sequence of 2 transformations. Select a transformation from each set of choices to make the statement true. Figure 1 can be transformed onto figure 3 by:

A a reflection across the y-axis B a rotation 90º clockwise about the origin C a translation 7 units to the right D a reflection across the x-axis E a rotation 180º clockwise about the origin F a translation 3 units to the left

followed by

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23 Part A

What are the signs of the coordinates (x, y) of point P'? A Both x and y are positive B x is negative and y is positive C Both x and y are negative D x is positive and y is negative Triangle PQR is shown on the coordinate plane. Triangle PQR is rotated 90º counterclockwise about the origin to form the image of triangle P'Q'R' (not shown). Then triangle P'Q'R' is reflected across the x-axis to form triangle P"Q"R" (not shown).

From PARCC PBA sample test non-calculator #6

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24 Part B

What are the signs of the coordinates (x, y) of point Q''? A Both x and y are positive B x is negative and y is positive C Both x and y are negative D x is positive and y is negative Triangle PQR is shown on the coordinate plane. Triangle PQR is rotated 90º counterclockwise about the origin to form the image of triangle P'Q'R' (not shown). Then triangle P'Q'R' is reflected across the x-axis to form triangle P"Q"R" (not shown).

From PARCC PBA sample test non-calculator #6

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Dilations

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Slide 72 / 230 Dialations

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A dilation is a transformation in which a figure is enlarged or

Dilation

reduced around a center point using a scale factor # 0. The center point is not altered. Note: This is the one transformation that does not usually result in congruent figures.

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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.

Dilation

When the scale factor of a dilation is less than 1, but greater than 0, the dilation is a reduction. When the scale factor is |1|, the dilation is an identity. This is the

• ne case when a dilation results in congruent figures.

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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?

Dilation Slide 76 / 230

x y

A A' B B' C C' D D' 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 center for this dilation was the origin (0,0).

Dialation Slide 77 / 230

x y 25 What is the scale factor for the image shown below? The pre-image is dotted and the image is solid. A 2 B 3 C

• 3

D 4

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26 What 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)

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27 What 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)

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28 What 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)

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29 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

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30 The 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

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31 The 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

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32 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

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33 Which of the following figures represents a reflection? A Figure A B Figure B C Figure C D Figure D

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34 Which of the following figures represents a dilation? A Figure A B Figure B C Figure C D Figure D

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35 Which of the following figures represents a translation? A Figure A B Figure B C Figure C D Figure D

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Symmetry

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Slide 89 / 230 Symmetry Slide 90 / 230 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.

Symmetry Slide 92 / 230

Do these images have symmetry? Where?

Symmetry Slide 93 / 230

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.

Asymmetrical Slide 94 / 230

Click the picture below to learn how to make your own face symmetrical. Tina Fey

Symmetry Slide 95 / 230

Rotational symmetry is when a figure can be rotated around a point onto itself using a turn that is less than 360º. Rotate the figure below to see the amount of times that the figure maps onto itself.

Symmetry Slide 96 / 230 Symmetry

To determine the degrees of each rotational symmetry :

• 1. Divide 360° by the number of times that the figure maps
• nto itself.
• 2. Keep adding that number until you reach a number that is

greater than or equal to 360 °. Note: the number greater than or equal to 360 ° does not count. Degrees of symmetry = 60 °, 120°, 180°, 240°, 300° 360 6 = 60°

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Rotational symmetry is when a figure can be rotated around a point onto itself in less than a 360 º turn. Rotate these figures. What degree of rotational symmetry do each of these figures have?

Symmetry Slide 98 / 230

36 How many lines of symmetry does this figure have? A 3 B 6 C 5 D 4

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37 How many lines of symmetry does this figure have? A 3 B 6 C 5 D 4

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38 Which figure's dotted line shows a line of symmetry? A B C D

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39 Which of the object does not have rotational symmetry? A B C D

Rotational symmetry is when a figure can be rotated around a point onto itself in less than a 360° turn.

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40 Determine the degrees of the rotational symmetry in the figure below. A B C D

Remember: divide 360° by the number of times that the object is rotationally symmetric

Click for hint.

90° 180° 120° 270°

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41 Determine the degrees of the rotational symmetry in the figure below. Choose all that apply. A 60° B 90° C 120° D 180° E 240° F 300°

Remember: divide 360° by the number of times that the object is rotationally symmetric Click for hint.

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Congruence & Similarity

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Slide 105 / 230 Congruence and Similarity

Congruent shapes have the same size and shape. 2 figures are congruent if the second figure can be obtained from the first by a series of translations, reflections, and/or rotations. 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.

Congruence and Similarity Slide 107 / 230

j° What would the value of j have to be in order for the figures below to be similar? 180 - 112 - 33 = 35 j = 35

Similarity Slide 108 / 230

Slide 109 / 230

42 Which pair of shapes is similar but not congruent? A B C D

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43 Which pair of shapes is similar but not congruent? A B C D

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44 Which of the following terms best describes the pair of figures? A congruent B similar C neither congruent nor similar

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45 Which of the following terms best describes the pair of figures? A congruent B similar C neither congruent nor similar

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46 Which of the following terms best describes the 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 pair of figures? A congruent B similar C neither congruent nor similar

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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.

Congruent vs. Similar

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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.

Congruent vs. Similar Slide 116 / 230

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.

Click on the location

• f the middle figure

to have it appear, if needed.

Congruent vs. Similar Slide 117 / 230

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.

Click on the location

• f the middle figure

to have it appear, if needed.

Congruent vs. Similar Slide 118 / 230

47 Which of these segments could be the image of segment AB after a sequence of reflections, rotations, and/or translations? Select each correct answer. A line segment CD B line segment EF C line segment GH D line segment JK E line segment LM F line segment NP

From PARCC PBA sample test non-calculator #5

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48 Part A Describe a single transformation that shows that triangle A'B'C' is congruent to triangle ABC. Include all the necessary information to complete the transformation. When you are finished, type in the number "1". In the coordinate plane shown, triangle ABC is congruent to triangle A'B'C'. Triangle A'B'C' is similar to triangle A"B"C".

From PARCC PBA sample test calculator #6

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49 Part B Describe a sequence of transformations that shows that triangle A"B"C" is similar to triangle A'B'C'. Include all the necessary information to complete the transformation. When you are finished, type in the number "1". In the coordinate plane shown, triangle ABC is congruent to triangle A'B'C'. Triangle A'B'C' is similar to triangle A"B"C".

From PARCC PBA sample test calculator #6

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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.

Special Pairs of Angles

40º 50º 40º 140º

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Vertical Angles are two angles that are opposite each

• ther when two lines intersect.

1 2 3 4 In this example, the vertical angles are: Vertical angles have the same measurement. So: ∠1 & ∠3 ∠2 & ∠4 m∠1 = m∠3 m∠2 = m∠4

Special Pairs of Angles Slide 124 / 230

x 2 4 1 3

Transformations

Line x cuts angles 1 and 3 in half. When angle 2 is reflected over line x, it forms angle 4. When angle 4 is reflected over line x, it forms angle 2.

∠2 ≅ ∠4 ∠4 ≅ ∠2

Vertical Angles can further be explained using the transformation of reflections.

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y 1 2 4 3 Line y cuts angles 2 and 4 in half. When angle 1 is reflected over line y, it forms angle 3. When angle 3 is reflected over line y, it forms angle 1.

Transformations

∠1 ≅ ∠3 ∠3 ≅ ∠1

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m∠2 = 40° m∠1 = 180 - 40 m∠1 = 140° m∠3 = 180 - 40 m∠3 = 140° 2 3 1 Using what you know about complementary, supplementary and vertical angles, find the measure of each missing angle.

By Vertical Angles: By Supplementary Angles:

Click Click

Transformations

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50 Are angles 2 and 4 vertical angles? Yes No 1 2 3 4

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51 Are angles 2 and 3 vertical angles? Yes No 1 2 3 4

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52 If angle 1 is 60º, what is the measure of angle 3? You must be able to explain why. 2 1 3 4 A 30º B 60º C 120º D 15º

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53 If angle 1 is 60º, what is the measure of angle 2? You must be able to explain why. 2 1 3 4 A 30º B 60º C 120º D 15º

Slide 131 / 230

A B C D ∠ABC is adjacent to ∠CBD How do you know? · They have a common side (ray ) · They have a common vertex (point B) 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.

Adjacent Angles Slide 132 / 230 Adjacent or Not Adjacent?

Explain why the angles are or are not adjacent. a b a b a b Adjacent The angles share a common side & vertex Not Adjacent The angles do not share a common side nor a common vertex Not Adjacent The angles do not share a common vertex

click to reveal click to reveal click to reveal

Slide 133 / 230

54 Which two angles are adjacent to each other? A 1 and 4 B 2 and 4 1 2 3 4 5 6

Slide 134 / 230

55 Which two angles are adjacent to each other? A 3 and 6 B 5 and 4 1 2 3 4 5 6

Slide 135 / 230

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.

Transversal Slide 136 / 230 Recall From 3rd Grade

Shapes and Perimeters Parallel lines are a set of two lines in the same plane that do not intersect (touch).

Slide 137 / 230

Corresponding Angles are on the same side of the transversal and in the same location at each intersection.

1 2 8 3 7

4

6 5

T r a n s v e r s a l

In this diagram the corresponding angles are:

Corresponding Angles Slide 138 / 230

56 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

Slide 139 / 230

57 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

Slide 140 / 230

58 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

Slide 141 / 230

59 Which are pairs of corresponding angles? A 2 and 4 B 6 and 5 C 7 and 8 D 1 and 3

1 2 3 4 5 6 7 8

Slide 142 / 230

Alternate Exterior Angles are on opposite sides of the transversal and on the outside of the given lines.

1 2 8 3 7

4

6 5

k m n In this diagram the alternate exterior angles are: Which line is the transversal?

Alternate Exterior Angles Slide 143 / 230

Alternate Interior Angles are on opposite sides of the transversal and on the inside of the given lines. In this diagram the alternate interior angles are:

1 2 8 3 7

4

6 5

k m n

Alternate Interior Angles Slide 144 / 230

Same Side Interior Angles are on same side of the transversal and on the inside of the given lines. In this diagram the same side interior angles are:

1 2 8 3 7

4

6 5

k m n

Same Side Interior Angles

Slide 145 / 230

60 Are angles 2 and 7 alternate exterior angles? Yes No 1 3 5 7 2 4 6 8 m n l

Slide 146 / 230

61 Are angles 3 and 6 alternate exterior angles? Yes No 1 3 5 7 2 4 6 8 m n l

Slide 147 / 230

62 Are angles 7 and 4 alternate exterior angles? Yes No 1 3 5 7 2 4 6 8 m n l

Slide 148 / 230

63 Which angle corresponds to angle 5? A B C D 1 3 5 7 2 4 6 8 m n l ∠3 ∠4 ∠2 ∠6

Slide 149 / 230

64 Which pair of angles are same side interior? A B C D 1 3 5 7 2 4 6 8 m n l

Slide 150 / 230

65 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 What type of angles are ∠3 and ∠6?

Slide 151 / 230

66 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 What type of angles are ∠5 and ∠2?

Slide 152 / 230

1 3 5 7 2 4 6 8 m n l

67 A Alternate Interior Angles B Alternate Exterior Angles C Corresponding Angles D Vertical Angles E Same Side Interior What type of angles are ∠5 and ∠6?

Slide 153 / 230

68 Are angles 5 and 2 alternate interior angles? Yes No 1 3 5 7 2 4 6 8 m n l

Slide 154 / 230

69 Are angles 5 and 7 alternate interior angles? Yes No 1 3 5 7 2 4 6 8 m n l

Slide 155 / 230

70 Are angles 7 and 2 alternate interior angles? Yes No 1 3 5 7 2 4 6 8 m n l

Slide 156 / 230

71 Are angles 3 and 6 alternate exterior angles? Yes No 1 3 5 7 2 4 6 8 m n l

Slide 157 / 230

1 3 5 7 2 4 6 8 k m n 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

Slide 158 / 230 Slide 159 / 230

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 160 / 230 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 161 / 230 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 162 / 230

4 5 6 2 7 1 8 k m n 72 Given the measure of one angle, find the measures of as many angles as possible. Which angles are congruent to the given angle? A B C D

Slide 163 / 230

4 5 6 2 7 1 8 k m n 73 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? A 50º B 40º C 130º

Slide 164 / 230

1 3 5 7 2 4 8 m n k 74 Given the measure of one angle, find the measures of as many angles as possible. Which angles are congruent to the given angle? A B C D

Slide 165 / 230

1 3 5 7 2 4 8 m n k 75 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? A 55º, 35º, 55º B 35º, 35º, 35º C 145º, 35º, 145º

Slide 166 / 230

76 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

Slide 167 / 230

1 3 5 7 2 4 6 8 b a t 77 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

Slide 168 / 230

1 3 5 7 2 4 6 8 b a t 78 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

Slide 169 / 230

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 ∆ABC B A C

Applying what we've learned to prove some interesting math facts... Slide 170 / 230

Let's draw a line through B parallel to AC. We then have 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°. k m n p B A C 2 1 k || m

Slide 171 / 230

m n p B A C 2 1 k k || m

• 1. ∠C ≅ ∠1 since if 2 parallel lines are cut by a transversal,

the alternate interior angles are congruent.

Slide 172 / 230

m n p B A C 2 1 k k || m

• 2. m∠2 = m∠B + m∠1 because if two parallel lines are cut by

a transversal, the alternate interior angles are congruent.

Slide 173 / 230

m n p B A C 2 1 k k || m

• 3. ∠A is supplementary with ∠2 because they are

supplementary angles that are adjacent.

Slide 174 / 230

• 4. Therefore,

m∠A + m∠2 = m∠A + m∠B + m∠1 = m∠A + m∠B + m∠C = 180°. m n p B A C 2 1 k k || m

Slide 175 / 230

Let's look at this another way...

• 1. ∠A ≅ ∠2 because if 2 parallel lines are cut by a

transversal, then alternate interior angles are congruent.

m n p B A C 1 2

k k || m

Slide 176 / 230

p B A C 1 2 m n

k k || m

• 2. ∠C ≅ ∠1 because if 2 parallel lines are cut by a

transversal, then alternate interior angles are congruent.

Slide 177 / 230

m n p B A C 1 2

k k || m

• 3. m ∠2 + m∠B + m∠1 = 180°, since all three angles form a

straight line.

Slide 178 / 230

m n p B A C 1 2

k k || m

• 4. Therefore,

m ∠2 + m∠B + m∠1 = m∠A + m∠B + m∠C = 180°.

Slide 179 / 230

Remote Exterior Angles

Return to Table of Contents

Slide 180 / 230

Exterior Angle Theorem - the measure of an exterior angle of a triangle is equal to the sum of the remote interior angles. B A C 1 Given ∆ABC

Exterior Angle Theorem

Exterior Angle Remote Interior Angles

Slide 181 / 230

We will use what we learned about special angles to see "why" and "how" the Remote Exterior Angle Theorem works and then we will practice applying this Theorem.

Exterior Angle Theorem Slide 182 / 230

Let's draw a line through B parallel to AC. We then have two parallel lines cut by a transversal. Number the angles and use what you know to prove the measure

• f m∠1 = sum of the measures of ∠B and ∠C.

m n p B A C 2 1 k k || m

Slide 183 / 230

m n p B A C 2 1 k k || m

• 1. ∠C ≅ ∠2 because if 2 parallel lines are cut by a

transversal, then alternate interior angles are congruent.

Slide 184 / 230

m n p B A C 2 1 k k || m

• 2. ∠1 = ∠B + ∠2 because if two parallel lines are cut by a

transversal, the alternate interior angles are congruent.

Slide 185 / 230

• 3. Therefore, m∠1 = m∠B + m∠2 = m∠B + m∠C.

m n p B A C 2 1 k k || m

Slide 186 / 230

Slide 187 / 230

2

Example

What is the measure of angle 2 in the diagram below? Diagram is NOT to scale. 163° = m∠2 + 27° m∠2 = 136°

Slide 188 / 230

3 What is the measure of angle 3 in the diagram below? Diagram is NOT to scale. 125° = m∠3 + 95° m∠3 = 30°

Slide 189 / 230

Find the value of x. Diagram is NOT to scale.

Slide 190 / 230 Slide 191 / 230

5

80 What is the measure of angle 5 in the diagram below?

Slide 192 / 230

6

81 What is the measure of angle 6 in the diagram below?

Slide 193 / 230

82 Find the value of x in the diagram below? Diagram is NOT to scale. (x + 5)° 36° (x - 7)°

Slide 194 / 230

83 What is the value of x in the diagram below? (2x - 3)° (3x)° 172°

Slide 195 / 230

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

Example

Name the pairs of angles whose sum is equal to m∠9.

Slide 196 / 230

p r

g h

1 2 3 4 5 6 7 8 9 10 11 12 13 14

84 Choose the expression that will make the statement below true: A B C D m∠12 = m∠1 + m∠6 m∠4 + m∠5 m∠5 + m∠6 m∠3 + m∠4

Slide 197 / 230 Slide 198 / 230

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

Example

What angles are congruent to angle 9?

Slide 199 / 230

86 Part A Explain why triangle RTS is similar to triangle VTU. When you finish writing down your answer, press the number "1" with your responder. The figure shows line RS parallel to line UV. The lines are intersected by 2 transversals. All lines are in the same plane. T S R V U

From PARCC PBA sample test calculator #5

Slide 200 / 230

87 Part B Given that m∠STV = 108º, determine m∠SRT + m∠TUV. Show your work or explain your answer. When you finish, enter the degree value that answers the question. The figure shows line RS parallel to line UV. The lines are intersected by 2 transversals. All lines are in the same plane. T S R V U

From PARCC PBA sample test calculator #5

Slide 201 / 230

Glossary & Standards

Return to Table of Contents

Slide 202 / 230 Standards for Mathematical Practices

Click on each standard to bring you to an example of how to meet this standard within the unit. MP8 Look for and express regularity in repeated reasoning. MP1 Make sense of problems and persevere in solving them. MP2 Reason abstractly and quantitatively. MP3 Construct viable arguments and critique the reasoning of others. MP4 Model with mathematics. MP5 Use appropriate tools strategically. MP6 Attend to precision. MP7 Look for and make use of structure.

Slide 203 / 230

Back to Instruction

Adjacent Angles

Two angles that are next to each other and have a common ray between them.

a b

a b

a b

Slide 204 / 230

Back to Instruction

Alternate Exterior Angles

When two lines are crossed by another line, the pairs of angles on opposite sides of the transversal but outside the two lines.

a b c d a b c d a b c d

Slide 205 / 230

Back to Instruction

Alternate Interior Angles

When two lines are crossed by another line, the pairs of angles on opposite sides of the transversal but inside the two lines.

a b c d a b c d a b c d

Slide 206 / 230

Back to Instruction

Asymmetrical

Something that is not symmetrical. Slide 207 / 230

Back to Instruction

Complimentary Angles

Two angles with a sum of 90 degrees. Slide 208 / 230

Back to Instruction

Congruent

Something that has the same size and shape. Two things that are equivalent.

segments shapes angles

Slide 209 / 230

Back to Instruction

Corresponding Angles

Angles that are on the same side of the transversal and in the same location at each intersection.

a a b b c c d d a a b b c c d d a a b b c c d d

Slide 210 / 230

shape remains the same!

dilation (enlargement)

Back to Instruction

Dilation

A transformation in which a figure is enlarged or reduced around a center point using a scale factor not equal to zero.

Each coordinate is multiplied by 2! A:(0,1) C:(3,0) B:(3,2) A':(0,2) C':(6,0) B':(6,4)

Slide 211 / 230

• S. F. = 2 > 1

3 = 2

( 6 )

Back to Instruction

Enlargement

A dilation where the scale factor is larger than one.

> 1

image is larger than pre- image Slide 212 / 230

• S. F. = 1 = 1

6 = 1

( 6 )

Back to Instruction

Identity

A dilation where the scale factor is the absolute value of one.

= 1

image is equal to pre-image Slide 213 / 230

after dilation after rotation after translation

Back to Instruction

Image

A figure that is composed after a transformation of a pre-image. Slide 214 / 230

Back to Instruction

Line of Symmetry

The imaginary line where you could fold the image and have both halves match exactly.

can be more than one!

Slide 215 / 230

Back to Instruction

Parallel Lines

A set of two lines in the same plane that do not intersect (touch). Slide 216 / 230

point outside figure

Back to Instruction

Point of Rotation

A point on a figure or some other point that a figure rotates (turns) around.

point on figure's edge point in middle

• f figure

Slide 217 / 230

before dilation before rotation before translation

Back to Instruction

Pre-Image

The original figure prior to a transformation. Slide 218 / 230

• S. F. = 1/2 < 1

=

( 3 )

1 6 2 Back to Instruction

Reduction

A dilation where the scale factor is less than one.

< 1

image is smaller than pre- image Slide 219 / 230

take note of reflection line! same distance to t

reflection (movement)

Back to Instruction

Reflection

A flip over a line that creates a mirror image of a figure, where each point in the image is the same distance from the line as the original point. Slide 220 / 230

r

• t

a t i

• n

( m

• v

e m e n t )

Back to Instruction

Rotation

A turn that moves a figure around a point.

Label by: and point

• f rotation

direction This figure is rotated 90o counter clockwise about point A.

Slide 221 / 230

Back to Instruction

Rotational Symmetry

A transformation where a figure can be rotated around a point onto itself in less than a 360 degree turn. Slide 222 / 230

Back to Instruction

Same Side Interior Angles

When two lines are crossed by a transversal, the pairs of angles on the same side of the transversal but inside the two lines.

a b c d a b c d a b c d

Slide 223 / 230

Scale Factor = 2

3 6 = 2)

(

Back to Instruction

Scale Factor

The ratio of the sides in an image to the sides in a pre image.

= 0

Slide 224 / 230

Back to Instruction

Similar

Two things that have the same shape, congruent angles, and proportional sides. congruent

special case of similarity when the sides form a proportion of 1.

Slide 225 / 230

Back to Instruction

Supplementary Angles

Two angles with a sum of 180 degrees.

S

Way to Remember: By drawing the extra line w/ the "S", you form an 8, for 180°

Slide 226 / 230

dilation (enlargement) r

• t

a t i

• n

( m

• v

e m e n t ) t r a n s l a t i

• n

( m

• v

e m e n t )

Back to Instruction

Transformation

Moving, enlarging, or shrinking a shape while maintaining the same angle measurements and proportional segment lengths. Slide 227 / 230

move to right 6 units move up 4 units

t r a n s l a t i

• n

( m

• v

e m e n t )

Back to Instruction

Translation

A slide that moves a figure to a different position (left, right, up, down) without changing its size

• r shape and without flipping or turning it.

state the rule: ( x + 6, y + 4 ) ( x + a, y + b )

Slide 228 / 230

Back to Instruction

Transversal

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

Slide 229 / 230

Also found in angles!

Back to Instruction

Vertex

Point where two or more straight lines/ faces/edges meet. A corner.

A C B

vertex vertex vertex

A triangle has 3 vertices.

Slide 230 / 230

Back to Instruction

Vertical Angles

Two angles that are opposite each other when two lines intersect.

70o 70o 110o 110o 120o 120o 60o X

x = 60o

Way to Remember: Vertical angles form 2 "V's" going in

• pposite directions