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Table of Contents
· 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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Any time you move, shrink, or enlarge a figure you make a transformation.
AB
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
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
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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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 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.
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?
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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AB
C
D
Translate pre-image ABCD 5 left and 3 up.
What are the coordinates of the image and pre-image?
Are the line segments in the pre-image and image the same length? In other words, was the size of the figure preserved?Both the pre-image and image are congruent.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.
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.· 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.
click
click
Translations Rule
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Write a rule for each translation.
2 Left and 5 UpA (3, -1) A' (1, 4)B (8, -1) B' (6, 4)C (7, -3) C' (5, 2)D (2, -4) D' (0, 1)
2 Left and 6 DownA (-2, 7) A' (-4, 1)B (-3, 1) B' (-5, -5)C (-6, 3) C' (-8, -3)
4 Right and 1 DownA (-5, 4) A' (-1, 3)B (-1, 2) B' (3, 1)C (-4, -2) C' (0, -3)D (-6, 1) D' (-2, 0)
5 Left and 3 UpA (3, 2) A' (-2, 5)B (7, 1) B' (2, 4)C (4, 0) C' (-1, 3)D (2, -2) D' (-3, 1)
(x, y) (x-2, y+5) (x, y) (x-2, y-6)
(x, y) (x-5, y+3) (x, y) (x+4, y-1)
click to reveal
click to revealclick to reveal
click to reveal
Translations Rule
click to reveal click to reveal
click to reveal click to reveal
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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'
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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DE
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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DE
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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DE
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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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
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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RotationsThe 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 counterclockwiseC 90 degreesD 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 counterclockwiseC 90 degreesD 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
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 90º counterclockwise.
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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A
B
C
DA'
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
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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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A B
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 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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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
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x
y
A B
CD
Reflect the figure across the y-axis.
Check to see if the pre-image and image are congruent.
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
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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.
Example
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x
y
A B
C D
EF
Reflect the figure across the y-axis.Click to see reflection.
Example
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x
y
AB
C
D
E
Reflect the figure across the line x = -2.Example
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x
y
A B
CD
Reflect the figure across the line y = x.
Example
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x
y
A
B C
A'
B' C'
15 The reflection below represents a reflection across:
A the x axisB 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 axisB 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 reflectionB rotation, 180º clockwiseC slideD rotation, 90º clockwise
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x
y
AB C
DE
A'
C'
B'
D'
E'
19 Describe the reflection below:
A across the line y = xB across the y axis
C across the line y = -3D across the x axis
Check to see if the pre-image and image are congruent.
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x
y
AB
C
A'
C'
B'
20 Describe the reflection below:
A across the line y = xB across the x-axis
C across the line y = -3D 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
23
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 originC 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 originC 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 AWhat are the signs of the coordinates (x, y) of point P'?A Both x and y are positiveB x is negative and y is positiveC Both x and y are negativeD 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 BWhat are the signs of the coordinates (x, y) of point Q''?
A Both x and y are positiveB x is negative and y is positiveC Both x and y are negativeD 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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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 one 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
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x
y
AA' 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
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x
y
25 What is the scale factor for the image shown below? The pre-image is dotted and the image is solid.
A 2B 3C -3D 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 3B -3C 1/3D -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 4B -4C 1/4D -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 3B -3.75C -3.5D 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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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
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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.
Asymmetrical
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Click the picture below to learn how to make your own face symmetrical.
Tina Fey
Symmetry
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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
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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°
Slide 97 / 230
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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36 How many lines of symmetry does this figure have?
A 3B 6C 5D 4
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37 How many lines of symmetry does this figure have?
A 3B 6C 5D 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 symmetricClick 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 symmetricClick for hint.
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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.
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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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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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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
Slide 111 / 230
44 Which of the following terms best describes the pair of figures?
A congruentB similarC neither congruent nor similar
Slide 112 / 230
45 Which of the following terms best describes the pair of figures?
A congruentB similarC neither congruent nor similar
Slide 113 / 230
46 Which of the following terms best describes the pair of figures?
A congruentB similarC 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 other 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 other 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 other through a series of translations, rotations, reflections and/or dilations. The pre-image is dotted, the image is solid.
Click on the locationof the middle figure to have it appear, if needed.
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 other through a series of translations, rotations, reflections and/or dilations. The pre-image is dotted, the image is solid.
Click on the locationof the middle figure to have it appear, if needed.
Congruent vs. Similar
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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 ADescribe 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 BDescribe 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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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 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
Special Pairs of Angles
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x2
41 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.
Slide 125 / 230
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
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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
Transformations
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50 Are angles 2 and 4 vertical angles?
YesNo
12
34
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51 Are angles 2 and 3 vertical angles?
YesNo
12
34
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52 If angle 1 is 60º, what is the measure of angle 3? You must be able to explain why.
21 3
4
A 30ºB 60ºC 120ºD 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.
21 34
A 30ºB 60ºC 120ºD 15º
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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 other and have a common ray between them. This means that they are on the same plane and they share no internal points.
Adjacent Angles
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Adjacent or Not Adjacent? Explain why the angles are or are not adjacent.
ab a
b
a
b
AdjacentThe angles share a common side & vertex
Not AdjacentThe angles do not share a common side nor a common vertex
Not AdjacentThe angles do not share a common vertex
click to reveal click to reveal click to reveal
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54 Which two angles are adjacent to each other?
A 1 and 4
B 2 and 4
1
23
456
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55 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.
Transversal
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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:
Corresponding Angles
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56 Which are pairs of corresponding angles?
A 2 and 6B 3 and 7C 1 and 8
1 23 4
5 67 8
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57 Which are pairs of corresponding angles?
A 2 and 6B 3 and 1C 1 and 8
1
23
45
6
78
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58 Which are pairs of corresponding angles?
A 1 and 5B 2 and 8C 4 and 8
1 2
3 4
5 6
7 8
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59 Which are pairs of corresponding angles?
A 2 and 4
B 6 and 5
C 7 and 8
D 1 and 3
1
23
4 5
6
7
8
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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?
Alternate Exterior Angles
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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:
12
8 3
7 4
6 5
k
m
n
Alternate Interior Angles
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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:
12
8 3
7 4
6 5
k
m
n
Same Side Interior Angles
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60 Are angles 2 and 7 alternate exterior angles?
YesNo
1 3
5 7
2 46 8
m
n
l
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61 Are angles 3 and 6 alternate exterior angles?
YesNo
1 3
5 7
2 46 8
m
n
l
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62 Are angles 7 and 4 alternate exterior angles?
YesNo
1 3
5 7
2 46 8
m
n
l
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63 Which angle corresponds to angle 5?
ABCD 1 3
5 7
2 46 8
m
n
l∠3∠4∠2∠6
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64 Which pair of angles are same side interior?
ABCD 1 3
5 7
2 46 8
m
n
l
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65
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?
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66
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 ∠5 and ∠2?
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1 3
5 7
2 46 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?
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68 Are angles 5 and 2 alternate interior angles?
YesNo
1 3
5 7
2 46 8
m
n
l
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69 Are angles 5 and 7 alternate interior angles?
YesNo
1 3
5 7
2 46 8
m
n
l
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70 Are angles 7 and 2 alternate interior angles?
YesNo
1 3
5 7
2 46 8
m
n
l
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71 Are angles 3 and 6 alternate exterior angles?
YesNo
1 3
5 7
2 46 8
m
n
l
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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 supplementaryare supplementary
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1 3
5 7
2 46 8
l
m
n
d
c
Reflections Continued
Line d cuts angles 2 and 8 in half.
When angle 4 is reflected over line d, it forms angle 6.
When angle 6 is reflected over line d, it forms angle 4.
Line c cuts angles 1 and 7 in half.
When angle 3 is reflected over line c, it forms angle 5.
When angle 5 is reflected over line c, it forms angle 3.
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Translations1 3
5 7
m
2 46 8
l
n
Line m is parallel to line l.
If line m is translated y units down, it will overlap with line l.
2 46 8
l
n
1 3
5 7
m
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Translations Continued
If line m is then translated x units left, all angles formed by lines m and n will overlap with all angles formed by lines l and n.2 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
m2 46 8
l
n
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4 56
2 71 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
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4 56
2 71 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 50ºB 40º C 130º
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1 35 7
2 48
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
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1 35 7
2 48
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, 35ºC 145º, 35º, 145º
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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
13
57
24
68
b
a
t
Slide 167 / 230
13
57
24
68
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
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13
57
24
68
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
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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...
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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
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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.
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mn
p
B
A C2
1k
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.
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mn
p
B
A C2
1 k
k || m
3. ∠A is supplementary with ∠2 because they are supplementary angles that are adjacent.
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4. Therefore, m∠A + m∠2 = m∠A + m∠B + m∠1 = m∠A + m∠B + m∠C = 180°.
mn
p
B
A C2
1 k
k || m
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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
12k
k || m
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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.
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m
n p
B
A C
12 k
k || m
3. m ∠2 + m∠B + m∠1 = 180°, since all three angles form a straight line.
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m
n p
B
A C
12k
k || m
4. Therefore, m ∠2 + m∠B + m∠1 = m∠A + m∠B + m∠C = 180°.
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Exterior Angle Theorem - the measure of an exterior angle of a triangle is equal to the sum of the remote interior angles.
B
AC1
Given ∆ABC
Exterior Angle Theorem
Exterior Angle
Remote Interior Angles
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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
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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 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.
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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.
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3. Therefore, m∠1 = m∠B + m∠2 = m∠B + m∠C .
m
n p
B
A C
2
1
k
k || m
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2
ExampleWhat is the measure of angle 2 in the diagram below? Diagram is NOT to scale.
163° = m∠2 + 27°m∠2 = 136°
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3
What is the measure of angle 3 in the diagram below? Diagram is NOT to scale.
125° = m∠3 + 95°m∠3 = 30°
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Find the value of x. Diagram is NOT to scale.
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5
80 What is the measure of angle 5 in the diagram below?
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6
81 What is the measure of angle 6 in the diagram below?
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82 Find the value of x in the diagram below? Diagram is NOT to scale.
(x + 5)°
36°(x - 7)°
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83 What is the value of x in the diagram below?
(2x - 3)°
(3x)°
172°
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p
r
g h
1 2 3
456
7 8910
11 121314
ExampleName the pairs of angles whose sum is equal to m∠9.
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p
r
g h
1 2 3
456
7 8910
11 121314
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
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p
r
g h
1 2 3
456
7 8910
11 121314
ExampleWhat angles are congruent to angle 9?
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86 Part AExplain 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
SR
VU
From PARCC PBA sample test calculator #5
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87 Part BGiven 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
SR
VU
From PARCC PBA sample test calculator #5
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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.
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Back to
Instruction
Adjacent Angles
Two angles that are next to each other and have a common ray between them.
a
b ab
a b
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Back to
Instruction
Alternate Exterior AnglesWhen 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
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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
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Instruction
Complimentary Angles
Two angles with a sum of 90 degrees.
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Instruction
Congruent
Something that has the same size and shape.Two things that are equivalent.
segments
shapes
angles
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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
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shape remains the same!
dilation(enlargement)
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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)
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S. F. = 2 > 1
3 = 2( 6 )
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Instruction
Enlargement
A dilation where the scale factor is larger than one.
> 1
image is larger
than pre-image
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S. F. = 1 = 1
6 = 1( 6 )
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Instruction
Identity
A dilation where the scale factor is the absolute value of one.
= 1image is equal to
pre-image
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after dilationafter rotation
after translation
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Instruction
Image
A figure that is composed after a transformation of a pre-image.
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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!
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Instruction
Parallel Lines
A set of two lines in the same plane that do not intersect (touch).
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point outside figure
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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 of figure
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before dilation
before rotation
before translation
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Instruction
Pre-Image
The original figure prior to a transformation.
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S. F. = 1/2 < 1
= ( 3 )16 2
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Instruction
Reduction
A dilation where the scale factor is less than one.
< 1
image is smaller than pre-
image
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take note of reflection line!
same distance to t
reflection(movement)
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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.
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rotation
(movement)
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Instruction
Rotation
A turn that moves a figure around a point.
Label by:
and point of rotation
direction
This figure is rotated
90o counter
clockwise about
point A.
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Instruction
Rotational SymmetryA transformation where a figure can be rotated around a point onto itself in less
than a 360 degree turn.
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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
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Scale Factor = 2
36
= 2)(
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Instruction
Scale Factor
The ratio of the sides in an image to the sides in a pre image.
= 0
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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.
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Instruction
Supplementary Angles
Two angles with a sum of 180 degrees.
SWay to
Remember:
By drawing the extraline w/ the "S", you
form an 8, for 180°
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dilation(enlargement)rotation
(movement)translation
(movement)
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Instruction
TransformationMoving, enlarging, or shrinking a shape while maintaining the same angle measurements
and proportional segment lengths.
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move to right 6 units
move up 4 units
translation
(movement)
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Instruction
Translation
A 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.
state the rule:
( x + 6, y + 4 )
( x + a, y + b )
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Instruction
Transversal
A line that cuts across two or more (usually parallel) lines.
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Also found
in angles!
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Instruction
Vertex
Point where two or more straight lines/faces/edges meet. A corner.
A
CB
vertex
vertexvertex
A triangle has 3
vertices.
Slide 230 / 230
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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
