Counterexamples 4 3 2 1 Factor...Translations Rule Slide 23 / 227 2 units Left … x-coordinate - 2 y-coordinate stays rule = (x - 2, y) 5 units Right & 3 units Down… x-coordinate

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8th Grade Math

2D Geometry: Transformations

www.njctl.org

2015-01-15

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Links to PARCC sample questions

Non-Calculator #8

Non- Calculator #12

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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

· Glossary· Remote Exterior Angles

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Sometimes when you subtract the fractions, you find that you can't because the first numerator is smaller than the second! When this happens, you need to regroup from the whole number.

How many thirds are in 1 whole?

How many fifths are in 1 whole?

How many ninths are in 1 whole?

Vocabulary words are identified with a dotted underline.

The underline is linked to the glossary at the end of the Notebook. It can also be printed for a word wall.

(Click on the dotted underline.)

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Back to

Instruction

FactorA whole number that can divide into another number with no remainder.

15 3 5

3 is a factor of 153 x 5 = 15

3 and 5 are factors of 15

1635 .1R

3 is not a factor of 16

A whole number that multiplies with another number to make a third number.

The charts have 4 parts.

Vocab Word1

Its meaning 2

Examples/ Counterexamples

3Link to return to the instructional page.

4

(As it is used in the

lesson.)

http://www.njctl.org




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

AB

C

A'B'

C'

pre-image image

Transformation

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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

Transformation

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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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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.

Translation

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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

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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

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?

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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


Slide 18 / 227Translate pre-image ABCD 4 right and 1 down. What are the coordinates of the image and pre-image?

A

B

C

D


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

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

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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?


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 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)

Translations Rule

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Translating left/right changes the x-coordinate.

Translating up/down changes the y-coordinate.





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

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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)

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

Translations Rule

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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)

Translations Rule

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DE

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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DE

F

G

D'E'

F'

G'


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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DE

F

G

D'E'

F'

G'


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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DE

F

G

D'E'

F'

G'


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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DE

F

G

D'E'

F'

G'


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

P

Rotations

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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.

A

This figure is rotated 90º counterclockwise

about point A.

B

This figure is rotated 180º clockwise about point B.

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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 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 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?

Rotations

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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.

Rotations

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A, A'C'

C

BB'

D'E'

D

E

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


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A B

CD

A'B'

C' D'

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


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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?

Rotations

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A B

CD

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

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Summarize what happens to the coordinates during a rotation? 90º Counterclockwise:

Half-turn:

90º Clockwise:

Rotations

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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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AB

CD

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 sample test

http://practice.parcc.testnav.com/#

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Reflections


Slide 50 / 227Examples

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

Reflection

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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 ABC is the same distance from line t as the vertices in A'B'C'.


Reflection

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x

y

A B

CD

Reflect the figure across the y-axis.


Reflection

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x

y

A B

CD

A'B'

C'D'

What do you notice about the coordinates when you reflect across the y-axis?

Reflection

page1svg

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x

y

A B

CD

A' B'

C'D'

What do you predict about the coordinates when you reflect across the x-axis?

Reflection

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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

AB

C

D

E

Reflect the figure across the line x = -2.

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x

y

A B

CD

Reflect the figure across the line y = x.

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x

y

A

B C

A'

B' C'

14 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

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x

y

D

B C

A

A'

C' B'

D'

15 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

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16 Which of the following represents a single reflection of Figure 1?

A

B

C

D

Figure 1

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17 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

DE

A'

C'

B'

D'

E'

18 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


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x

y

A

B

C

A'

C'

B'

19 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


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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 sample test

1

23

y

x

http://practice.parcc.testnav.com/#

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20 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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21 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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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.

Dilation

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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, but greater than 0, the dilation is a reduction.

When the scale factor is |1|, the dilation is an identity.

Dilation

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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

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x

y

AA' B

B'

C C'DD'

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).

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x

y

22 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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23 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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24 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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25 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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26 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

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27 The coordinates of a point change as follows during a dilation:

(4, -9) (16, -36)


A 4B -4

C 1/4

D -1/4

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28 The 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

Slide 82 / 22729 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

Slide 83 / 22730 Which of the following figures represents a reflection?

A Figure A B Figure B


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31 Which of the following figures represents a dilation?

A Figure A B Figure B


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


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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.

Symmetry

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Do these images have symmetry? Where?

Symmetry

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.

Symmetry

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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 360º turn.

Rotate the figure below to see the amount of times that the figure maps onto itself.

Symmetry

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SymmetryTo determine the degrees of each rotational symmetry: 1. Divide 360° by the number of times that the figure maps onto 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 360º turn.

Rotate these figures. What degree of rotational symmetry do each of these figures have?

Symmetry

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33 How many lines of symmetry does this figure have?

A 3

B 6

C 5

D 4

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

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

A B C D

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35 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.Click for hint.

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36 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 symmetricClick for hint.

90°

180°

120°

270°

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37 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 symmetricClick 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 obtained from the first by a series of translations, reflections, and/or rotations.

Remember - translations, reflections and rotations preserve image size and shape.

page1svg

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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

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Click for web page

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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

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38 Which pair of shapes is similar but not congruent?

A

B

C

D

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39 Which pair of shapes is similar but not congruent?

A

B

C

D

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

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

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40 Which of the following terms best describes the pair of figures?

A congruent

B similar

C neither congruent nor similar

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41 Which of the following terms best describes the pair of figures?

A congruent

B similar


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42 Which of the following terms best describes the ##################### pair of figures?

A congruent

B 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 other through a series of translations, rotations, reflections and/or dilations. The pre-image is dotted, the image is solid.

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Click on the locationof the middle figure to have it appear, if needed.

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Click on the locationof the middle figure to have it appear, if needed.

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Special Pairs of Angles


Slide 117 / 227Recall:

· Complementary Angles are two angles with a sum of 90 degrees.

These two angles are complementary angles because their sum is 90.

Notice that they form a right angle when placed together.

· Supplementary Angles are two angles with a sum of 180 degrees.

These two angles are supplementary angles because their sum is 180.

Notice that they form a straight angle when placed together.

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Vertical Angles are two angles that are opposite each other when two lines intersect.

12

34

In this example, the vertical angles are:

Vertical angles have the same measurement. So:

∠1 & ∠3∠2 & ∠4

m∠1 = m∠3m∠2 = m∠4

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x2

41 3

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

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

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y

12

43

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

page1svg

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m∠2 = 40° m∠1 = 180 - 40m∠1 = 140°

m∠3 = 180 - 40m∠3 = 140°

23

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

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43 Are angles 2 and 4 vertical angles?

Yes

No

12

34

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44 Are angles 2 and 3 vertical angles?

Yes

No

12

34

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45 If angle 1 is 60º, what is the measure of angle 3? You must be able to explain why.

21 3

4

A 30º

B 60º

C 120º

D 15º

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46 If angle 1 is 60º, what is the measure of angle 2? You must be able to explain why.

21

34

A 30º

B 60º

C 120º

D 15º

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A

B

C

D

is adjacent to

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 other and have a common ray between them. This means that they are on the same plane and they share no internal points.

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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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47 Which two angles are adjacent to each other?

A 1 and 4

B 2 and 4

1

23

456

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48 Which two angles are adjacent to each other?

A 3 and 6

B 5 and 4

12

34 5

6

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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 in the same plane that do not intersect (touch).

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

1 28 3

7 4

6 5

Tran

sver

sal

In this diagram the corresponding angles are:

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

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

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49 Which are pairs of corresponding angles?

A 2 and 6

B 3 and 7

C 1 and 81 2

3 45 6

7 8

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A 2 and 6

B 3 and 1

C 1 and 8

1

23

45

6

78

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A 1 and 5

B 2 and 8

C 4 and 8

1 2

3 4

5 6

7 8

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A 2 and 4

B 6 and 5

C 7 and 8

D 1 and 3

1

23

4 5

6

7

8

Slide 137 / 227

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

12

8 3

7 4

6 5

k

m

n

In this diagram the alternate exterior angles are:

Which line is the transversal?

Slide 138 / 227

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

28 3

7 4

6 5

k

m

n

Slide 139 / 227

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

28 3

7 4

6 5

k

m

n

Slide 140 / 227

53 Are angles 2 and 7 alternate exterior angles?

Yes

No1 3

5 7

2 46 8

m

n

l

Slide 141 / 227


Yes

No

1 3

5 7

2 46 8

m

n

l

Slide 142 / 227


Yes

No

1 3

5 7

2 46 8

m

n

l

Slide 143 / 227

56 Which angle corresponds to angle 5?

AB

C

D1 3

5 7

2 46 8

m

n

l

Slide 144 / 227

57 Which pair of angles are same side interior?

AB

C

D1 3

5 7

2 46 8

m

n

l

Slide 145 / 227

58

A Alternate Interior Angles

B Alternate Exterior Angles

C Corresponding Angles

D Vertical Angles

1 3

5 7

2 46 8

m

n

l

E Same Side Interior

What type of angles are ∠3 and ∠6?

Slide 146 / 227

59




D Vertical Angles

1 3

5 7

2 46 8

m

n

l



Slide 147 / 227

60




D Vertical Angles

1 3

5 7

2 46 8

m

n

l



Slide 148 / 227

61 Are angles 5 and 2 alternate interior angles?

Yes

No

1 3

5 7

2 46 8

m

n

l

Slide 149 / 227


Yes

No

1 3

5 7

2 46 8

m

n

l

Slide 150 / 227


Yes

No

1 3

5 7

2 46 8

m

n

l

Slide 151 / 227


Yes

No

1 3

5 7

2 46 8

m

n

l

Slide 152 / 227

1 35 7

2 46 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 153 / 227 Slide 154 / 227

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 155 / 227

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 156 / 227

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 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 157 / 227

4 56

2 71 8

k

m

n

65 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 158 / 227

4 56

2 71 8

k

m

n

66 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 159 / 227

1 3

5 7

2 48

m

n

k

67 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 160 / 227

1 3

5 7

2 48

m

k

68 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?

n

A 55º, 35º, 55º

B 35º, 35º, 35º

C 145º, 35º, 145º

Slide 161 / 227

69 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

13

57

24

68

b

a

t

Slide 162 / 227


A Reflection Only

B Translation Only



13

57

24

68

b

a

t

Slide 163 / 227


A Reflection Only

B Translation Only



13

57

24

68

b

a

t

Slide 164 / 227

Applying what we've learned to prove some interesting math facts...

Slide 165 / 227

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 166 / 227

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 C2

1

k || m

Slide 167 / 227

mn

p

B

A C2

1 k

k || m

1. ∠C ≅ ∠1 since if 2 parallel lines are cut by a transversal, the alternate interior angles are congruent.

Slide 168 / 227

mn

p

B

A C2

1 k

k || m

2. ∠2 = ∠B + ∠1 because if two parallel lines are cut by a transversal, the alternating exterior angles are congruent.

Slide 169 / 227

mn

p

B

A C2

1 k

k || m

3. ∠A is supplementary with ∠2 because if 2 parallel lines are cut by a transversal, then same side interior angles are supplementary.

Slide 170 / 227

4. Therefore, ∠A + ∠2 = ∠A + ∠B + ∠1 = ∠A + ∠B + ∠C = 180°.

mn

p

B

A C2

1 k

k || m

Slide 171 / 227

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

12 k

k || m

Slide 172 / 227

p

B

A C

12

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 173 / 227

m

n p

B

A C

12 k

k || m

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

Slide 174 / 227

m

n p

B

A C

12k

k || m

4. Therefore, ∠2 + ∠B + ∠1 = ∠A + ∠B + ∠C = 180°.

Slide 175 / 227

Remote Exterior Angles


Slide 176 / 227 Exterior Angle Theorem - the measure of an exterior angle of a triangle is equal to the sum of the remote interior angles.

B

A C1

Exterior Angle

Remote Interior Angles

Given

Slide 177 / 227

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.

Slide 178 / 227

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 of ∠1 = sum of the measures of ∠B and ∠C.

m

n p

B

A C

2

1

k

k || m

Slide 179 / 227

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 180 / 227

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 alternating exterior angles are congruent.

page1svg

Slide 181 / 227

3. Therefore, ∠1 = ∠B + ∠2 = ∠B + ∠C.

m

n p

B

A C

2

1

k

k || m

Slide 182 / 227

Slide 183 / 227

2

ExampleWhat is the measure of angle 2 in the diagram below? Diagram is NOT to scale.

163° = m∠2 + 27°m∠2 = 136°

Slide 184 / 227

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 185 / 227

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

Slide 186 / 227

Slide 187 / 227

5

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

Slide 188 / 227

6

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

Slide 189 / 227

75 Find the value of x in the diagram below? Diagram is NOT to scale.

(x + 5)°

(10x - 34)°(x - 7)°

Slide 190 / 227

76 What is the value of x in the diagram below?

(2x - 3)°

(3x)°

172°

Slide 191 / 227

p

r

g h

1 2 3456

7 8910

11 121314

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

Slide 192 / 227

p

r

g h

1 2 3456

7 8910

11 121314

77 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 193 / 227 Slide 194 / 227

p

r

g h

1 2 3456

7 8910

11 121314

ExampleWhat angles are congruent to angle 9?

Slide 195 / 227

Glossary

Return toTable ofContents

Slide 196 / 227

Back to

Instruction

Adjacent Angles

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

ab a

ba b

Slide 197 / 227

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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 198 / 227

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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

page1svg

page117svg

page135svg

page136svg

Slide 199 / 227

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Instruction

Asymmetrical

Something that is not symmetrical.

Slide 200 / 227

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Instruction

Complimentary Angles

Two angles with a sum of 90 degrees.

=

90o

+45o

45o=

90o

+ 60o

30o C

Way to Remember:

By drawing the extraline w/ the "C", you

form a 9, for 90°

Slide 201 / 227

Back to

Instruction

CongruentSomething that has the same size and shape.

Two things that are equivalent.

segments

angles

shapes

30o

30o

Slide 202 / 227

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 203 / 227

dilation(enlargement)

Back to

Instruction

DilationA 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)

shape remains the same!

Slide 204 / 227

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Instruction

Enlargement

A dilation where the scale factor is larger than one.

> 1 image is larger

than pre-image

{ {36

S. F. = 2 > 1

3= 2( 6 )

page107svg

page95svg

page72svg

page128svg

page72svg

page141svg

Slide 205 / 227

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Instruction

IdentityA dilation where the scale factor is the

absolute value of one.

= 1image is equal to

pre-image

S. F. = 1 = 1

{

6

6= 1( 6 )

Slide 206 / 227

after

translationafter dilation

after rotationBack to

Instruction

ImageA figure that is composed after a

transformation of a pre-image.

Slide 207 / 227

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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 208 / 227

Back to

Instruction

Parallel Lines

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

Slide 209 / 227

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Instruction

Point of Rotation

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

point outside figurepoint in middle of figure

point on figure's edge

Slide 210 / 227

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Instruction

Pre-ImageThe original figure prior to a transformation.

before

translationbefore dilation

before

rotation

page141svg

page6svg

page41svg

page97svg

page8svg

page6svg

Slide 211 / 227

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Instruction

ReductionA dilation where the scale factor is less than one.

< 1 image is smaller than pre-

image

S. F. = 1/2 < 1

{ {36

6= ( 3 )2

1

Slide 212 / 227

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Instruction

ReflectionA 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.

reflection(movement)

{ 6 { 6same distance to t

{3 3{

6 6

{ {

take note of reflection line!

over line t

Slide 213 / 227

rotation

(movement)

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Instruction

RotationA turn that moves a figure around a point.

A

This figure is rotated

90o counter

clockwise about

point A.

Label by:

and point of rotation

directionA

Slide 214 / 227

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Instruction

Rotational Symmetry

A transformation where a figure can be rotated around a point onto itself in less than a 360 degree

turn.

90o

Slide 215 / 227

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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 216 / 227

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Instruction

Scale Factor

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

= 0

{ {36

Scale Factor = 2

36 = 2)(

page141svg

page72svg

page72svg

page42svg

page137svg

page22svg

Slide 217 / 227

Back to

Instruction

SimilarTwo 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 218 / 227

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Instruction

Supplementary Angles

Two angles with a sum of 180 degrees.

+

+180o

180o

=

=90o 90o

80o

100o

SWay to

Remember:

By drawing the extraline w/ the "S", you

form an 8, for 180°

Slide 219 / 227

Back to

Instruction

Transformation

Moving, enlarging, or shrinking a shape while maintaining the same angle measurements and

proportional segment lengths.

translation

(movement)

rotation

(movement)

dilation(enlargement)

Slide 220 / 227

translation

(movement)

Back to

Instruction

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

shape and without flipping or turning it.

move to right 6 units

move up 4 units

state the rule:

( x + 6, y + 4 ) ( x + a, y + b )

Slide 221 / 227

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Instruction

Transversal A line that cuts across two or more (usually

parallel) lines.

Slide 222 / 227

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Instruction

VertexPoint where two or more

straight lines/faces/edges meet.

A corner.

A

CBvertex

vertexvertex

A triangle has 3

vertices.

Also found

in angles!

page36svg

page95svg

page6svg

page72svg

page127svg

page119svg

Slide 223 / 227

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Instruction

Vertical Angles

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

70o

70o

110o110o 120o

120o

60oX

x = 60o

Way to Remember:

Vertical angles form 2 "V's" going in

opposite directions

Slide 224 / 227

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Instruction

Slide 225 / 227

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Instruction

Slide 226 / 227

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Instruction

Slide 227 / 227

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Instruction

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Documents

Counterexamples 4 3 2 1 Factor...Translations Rule Slide 23 / 227 2 units Left … x-coordinate - 2 y-coordinate stays rule = (x - 2, y) 5 units Right & 3 units Down… x-coordinate