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2D Geometry: Transformations 8th Grade Math 2D Geometry: Transformations 2013-03-06 www.njctl.org

Setting the PowerPoint View Use Normal View for the Interactive Elements To use the interactive elements in this presentation, do not select the Slide Show view. Instead, select Normal view and follow these steps to set the view as large as possible: On the View menu, select Normal. Close the Slides tab on the left. In the upper right corner next to the Help button, click the ^ to minimize the ribbon at the top of the screen.  On the View menu, confirm that Ruler is deselected. On the View tab, click Fit to Window. Use Slide Show View to Administer Assessment Items To administer the numbered assessment items in this presentation, use the Slide Show view. (See Slide 21 for an example.)

Table of Contents Transformations Translations Rotations Reflections Click on a topic to go to that section Translations Rotations Reflections Dilations Symmetry Congruence & Similarity Special Pairs of Angles Common Core Standards: 8.G.1, 8.G.2, 8.G.3, 8.G.4, 8.G.5

Transformations Return to Table of Contents

Any time you move, shrink, or enlarge a figure you make a transformation. If the figure you are moving (pre-image) is labeled with letters A, B, and C, you can label the points on the transformed image (image) with the same letters and the prime sign. B B' A A' Teacher Instructions: Triangle ABC is the pre-image to the reflected image A'B'C' pre-image image C C'

There are four types of transformations in this unit: Translations Rotations Reflections Dilations The first three transformations preserve the size and shape of the figure. In other words: If your pre-image is a trapezoid, your image is a congruent trapezoid. If your pre-image is an angle, your image is an angle with the same measure. If your pre-image contains parallel lines, your image contains parallel lines.

Translations Return to Table of Contents

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.

This shows a translation of pre-image ABC to image A'B'C' This shows a translation of pre-image ABC to image A'B'C'. Each point in the pre-image was moved right 7 and up 4.

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Both the pre-image and image are congruent. 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? A' B' D' C' Answer: A(3,-1) A’(1,4) B(8,-1) B’(6,4) C(7,-3) C’(5,2) D(2,-4) D’(0,1) 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.

Translate pre-image ABC 2 left and 6 down. What are the coordinates of the image and pre-image? A C B Answer: A(-2,7) A’(-4,1) B(-3,1) B’(-5,-5) C(-6,3) C’(-8,-3) 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.

Translate pre-image ABCD 4 right and 1 down. What are the coordinates of the image and pre-image? A B D Answer: A(-5,4) A’(-1,3) B(-1,2) B’(3,1) C(-4,-2) C’(0,-3) D(-6,1) D’(-2,0) 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.

Translate pre-image ABCD 5 left and 3 up. What is the rule and what are the new coordinates of the image. A B Answer: A(3,2) A’(-2,5) B(7,1) B’(2,4) C(4,0) C’(-1,3) D(2,-2) D’(-3,1) 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.

A rule can be written to describe translations on the coordinate plane A rule can be written to describe translations on the coordinate plane. Look at the following rules and coordinates to see if you can find a pattern. 2 Left and 5 Up A (3,-1) A' (1,4) B (8,-1) B' (6,4) C (7,-3) C' (5,2) D (2, -4) D' (0,1) 2 Left and 6 Down A (-2,7) A' (-4,1) B (-3,1) B' (-5,-5) C (-6,3) C' (-8,-3) 5 Left and 3 Up A (3,2) A' (-2,5) B (7,1) B' (2,4) C (4,0) C' (-1,3) D (2,-2) D' (-3,1) 4 Right and 1 Down A (-5,4) A' (-1,3) B (-1,2) B' (3,1) C (-4,-2) C' (0,-3) D (-6, 1) D' (-2,0)

Translating left/right changes the x-coordinate. Translating up/down changes the y-coordinate. 2 Left and 5 Up A (3,-1) A' (1,4) B (8,-1) B' (6,4) C (7,-3) C' (5,2) D (2, -4) D' (0,1) 2 Left and 6 Down A (-2,7) A' (-4,1) B (-3,1) B' (-5,-5) C (-6,3) C' (-8,-3) 5 Left and 3 Up A (3,2) A' (-2,5) B (7,1) B' (2,4) C (4,0) C' (-1,3) D (2,-2) D' (-3,1) 4 Right and 1 Down A (-5,4) A' (-1,3) B (-1,2) B' (3,1) C (-4,-2) C' (0,-3) D (-6, 1) D' (-2,0)

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

2 units Left … x-coordinate - 2 y-coordinate stays rule = (x - 2, y) A rule can be written to describe translations on the coordinate plane. 2 units Left … x-coordinate - 2 y-coordinate stays rule = (x - 2, y) 5 units Right & 3 units Down… x-coordinate + 5 y-coordinate - 3 rule = (x + 5, y - 3) click to reveal click to reveal

Write a rule for each translation. 2 Left and 5 Up A (3,-1) A' (1,4) B (8,-1) B' (6,4) C (7,-3) C' (5,2) D (2, -4) D' (0,1) 2 Left and 6 Down A (-2,7) A' (-4,1) B (-3,1) B' (-5,-5) C (-6,3) C' (-8,-3) click to reveal (x, y) (x-2, y+5) click to reveal (x, y) (x-2, y-6) 5 Left and 3 Up A (3,2) A' (-2,5) B (7,1) B' (2,4) C (4,0) C' (-1,3) D (2,-2) D' (-3,1) 4 Right and 1 Down A (-5,4) A' (-1,3) B (-1,2) B' (3,1) C (-4,-2) C' (0,-3) D (-6, 1) D' (-2,0) click to reveal (x, y) (x-5, y+3) click to reveal (x, y) (x+4, y-1)

What rule describes the translation shown? 1 What rule describes the translation shown? E' A (x,y) (x - 4, y - 6) D' F' B (x,y) (x - 6, y - 4) E (x,y) (x + 6, y + 4) D C F G' D (x,y) (x + 4, y + 6) Answer: C G

What rule describes the translation shown? 2 What rule describes the translation shown? A (x,y) (x, y - 9) E B (x,y) (x, y - 3) D C (x,y) (x - 9, y) F D (x,y) (x - 3, y) G Answer: A E' D' F' G'

What rule describes the translation shown? 3 What rule describes the translation shown? A (x,y) (x + 8, y - 5) B (x,y) (x - 5, y - 1) E' D' C (x,y) (x + 5, y - 8) F' D (x,y) (x - 8, y + 5) E D Answer: D G' F G

What rule describes the translation shown? 4 What rule describes the translation shown? A (x,y) (x - 3, y + 2) B (x,y) (x + 3, y - 2) E D F C (x,y) (x + 2, y - 3) E' D' D (x,y) (x - 2, y + 3) F' Answer: B G G'

What rule describes the translation shown? 5 What rule describes the translation shown? A (x,y) (x - 3, y + 2) E' B (x,y) (x + 3, y - 2) D' F' C (x,y) (x + 2, y - 3) E D (x,y) (x - 2, y + 3) D F G' Answer: D G

Rotations Return to Table of Contents

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

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

This figure is rotated 90 degrees counterclockwise about point A. 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. A B Teacher Instructions: Clockwise is to the right & Counter clockwise is to the left. Click for answer This figure is rotated 90 degrees counterclockwise about point A. Click for answer This figure is rotated 180 degrees clockwise about point B.

How is this figure rotated about the origin? C D A' B' C' D' How is this figure rotated about the origin? In a coordinate plane, each quadrant represents 90. This figure is rotated 270 degrees clockwise about the origin or 90 degrees counterclockwise about the origin. Click to Reveal Check to see if the pre-image and image are congruent.

The following descriptions describe the same rotation. What do you notice? Can you give your own example?

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.

6 How is this figure rotated about point A? (Choose more than one answer.) C D B A clockwise B' B counterclockwise E A, A' C' C 90 degrees Answer: B, C E' D 180 degrees D' E 270 degrees Check to see if the pre-image and image are congruent.

7 How is this figure rotated about point the origin? (Choose more than one answer.) A B C D A' B' C' D' A clockwise B counterclockwise C 90 degrees Answer: A, D D 180 degrees E 270 degrees Check to see if the pre-image and image are congruent.

What is the turn that will turn this hexagon onto itself? 8 What is the turn that will turn this hexagon onto itself? A B Answer: D C D

Write the coordinates for the pre-image and image. B C D A' B' C' D' Now let's look at the same figure and see what happens to the coordinates when we rotate a figure. Write the coordinates for the pre-image and image. What do you notice? When rotated 90 counter-clockwise, the x-coordinate is the opposite of the pre-image y-coordinate and the y-coordinate is the same as the pre-image of the x-coordinate. In other words: (x, y) (-y, x) Click to Reveal

What happens to the coordinates in a half-turn? Write the coordinates for the pre-image and image. What do you notice? A B C D A' B' C' D' Click to Reveal When rotated a half-turn, the x-coordinate is the opposite of the pre-image x-coordinate and the y-coordinate is the opposite of the pre-image of the y-coordinate. In other words: (x, y) (-x, -y)

Can you summarize what happens to the coordinates during a rotation? 90 Counterclockwise: Half-turn: 90 Clockwise: Click to Reveal (x, y) (-y, x) Click to Reveal (x, y) (-x, -y) Click to Reveal (x, y) (y, -x)

9 What are the new coordinates of a point A (5, -6) after a 90 rotation clockwise? A (-6, -5) B (-6, 5) C (-5, 6) D (5, -6) Answer: A

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

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

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

What are the new coordinates of a point Y (9, -12) after a Half-turn? 13 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) Answer: B

Reflections Return to Table of Contents

Example

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

A reflection is a flip because the figure is flipped over a line A reflection is a flip because the figure is flipped over a line. Each point in the image is the same distance from the line as the original point. A B C A' B' C' t A and A' are both 6 units from line t. B and B' are both 6 units from line t. C and C' are both 3 units from line t. Each vertex in ABC is the same distance from line t as the vertices in A'B'C'. Check to see if the pre-image and image are congruent.

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

What do you notice about the coordinates when you reflect across the y-axis? Tap box for coordinates A (-6, 5) A' (6, 5) B (-4, 5) B' (4, 5) C (-4, 1) C' (4, 1) D (-6, 3) D' (6, 3) When you reflect across the y-axis, the x-coordinate becomes the opposite. So (x, y) (-x, y) when you reflect across the y-axis. A B A' B' D D' C' C x Check to see if the pre-image and image are congruent.

What do you predict about the coordinates when you reflect across the x-axis? Tap box for coordinates A (-6, 5) A' (-6, -5) B (-4, 5) B' (-4, -5) C (-4, 1) C' (-4, -1) D (-6, 3) D' (-6, -3) When you reflect across the x-axis, the y-coordinate becomes the opposite. So (x, y) (x, -y) when you reflect across the x-axis. A B D C x C' D' A' B' Check to see if the pre-image and image are congruent.

Reflect the figure across the y-axis then the x-axis Reflect the figure across the y-axis then the x-axis. Click to see each reflection. Check to see if the pre-image and image are congruent.

Reflect the figure across the y-axis. Click to see reflection. Check to see if the pre-image and image are congruent.

Reflect the figure across the line x = -2. Check to see if the pre-image and image are congruent.

Reflect the figure across the line y = x. Check to see if the pre-image and image are congruent.

The reflection below represents a reflection across: 14 The reflection below represents a reflection across: A the x axis B the x axis, then the y axis C the y axis D the y axis, then the x axis y Answer: A A B C x B' C' A' Check to see if the pre-image and image are congruent.

The reflection below represents a reflection across: 15 The reflection below represents a reflection across: A the x axis B the x axis, then the y axis C the y axis D the y axis, then the y axis y A D Answer: C B C x C' B' D' A' Check to see if the pre-image and image are congruent.

Which of the following represents a single reflection of Figure 1? 16 Which of the following represents a single reflection of Figure 1? Figure 1 A C Answer: B B D

Which of the following describes the movement below? 17 Which of the following describes the movement below? A reflection B rotation, 90 clockwise C slide Answer: B D rotation, 180 clockwise

Describe the reflection below: 18 Describe the reflection below: A across the line y = x C across the line y = -3 B across the y axis D across the x axis y D' C' Answer: A E' B' B C A' x A E D Check to see if the pre-image and image are congruent.

Describe the reflection below: 19 Describe the reflection below: A across the line y = x B across the line y = -3 C across the x axis D across the line x = 4 y Answer: D A A' B B' x C C' Check to see if the pre-image and image are congruent.

Dilations Return to Table of Contents

A dilation is a transformation in which a figure is enlarged or reduced using a scale factor = 0. Its center is not altered.

The scale factor is the ratio of sides: When the scale factor of a dilation is greater than 1, the dilation is an enlargement. When the scale factor of a dilation is less than 1, the dilation is a reduction. When the scale factor is |1|, the dilation is an identity.

Example. If the pre-image is red and the image is blue, what type of dilation is this? What is the scale factor of the dilation? x y This is an enlargement. Scale Factor is 2.

What happened to the coordinates with a scale factor of 2? x y A (0, 1) A' (0, 2) B (3, 2) B' (6, 4) C (4, 0) C' (8, 0) D (1, 0) D' (2, 0) The coordinates were all multiplied by 2. B' A' B A D D' C C'

20 What is the scale factor for the image shown below? The pre-image is in red and the image is in blue. x y A 2 B 3 C -3 Answer: B D 4

21 What are the coordinates of a point S (3, -2) after a dilation with a scale factor of 4? A (12, -8) B (-12, -8) C (-12, 8) D (-3/4, 1/2) Answer: A

22 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) Answer: B

23 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) Answer: D

What is the scale factor? 24 The coordinates of a point change as follows during a dilation: (-6, 3) (-2, 1) What is the scale factor? A 3 B -3 Answer: C C 1/3 D -1/3

The coordinates of a point change as follows during a dilation: 25 The coordinates of a point change as follows during a dilation: (4, -9) (16, -36) What is the scale factor? A 4 B -4 Answer: A C 1/4 D -1/4

The coordinates of a point change as follows during a dilation: 26 The coordinates of a point change as follows during a dilation: (5, -2) (17.5, -7) What is the scale factor? A 3 B -3.75 Answer: D C -3.5 D 3.5

27 Which of the following figures represents a rotation? (and could not have been achieved only using a reflection) A Figure A B Figure B Answer: B C Figure C D Figure D

Which of the following figures represents a reflection? 28 Which of the following figures represents a reflection? A Figure A B Figure B Answer: A C Figure C D Figure D

Which of the following figures represents a dilation? 29 Which of the following figures represents a dilation? A Figure A B Figure B Answer: D C Figure C D Figure D

Which of the following figures represents a translation? 30 Which of the following figures represents a translation? A Figure A B Figure B Answer: C C Figure C D Figure D

Symmetry Return to Table of Contents

Symmetry A line of symmetry divides a figure into two parts that match each other exactly when you fold along the dotted line. Draw the lines of symmetry for each figure below if they exist.

Which of these figures have symmetry? Draw the lines of symmetry.

Do these images have symmetry? Where?

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

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

How many lines of symmetry does this figure have? 31 How many lines of symmetry does this figure have? A 3 B 6 C 5 D 4 Answer: B

Which figure's dotted line shows a line of symmetry? 32 Which figure's dotted line shows a line of symmetry? A B C D Answer: C

Which of the objects does not have rotational symmetry? 33 Which of the objects does not have rotational symmetry? A B Answer: D C D Click for hint. Rotational symmetry is when a figure can be rotated around a point onto itself in less than a 360 turn.

Congruence & Similarity Return to Table of Contents

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.

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. Teacher Instructions: Notice a dilation is only used to create similar figures NOT congruent

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Click to reveal

Which pair of shapes is similar but not congruent? 34 Which pair of shapes is similar but not congruent? A B Answer: D C D

Which pair of shapes is similar but not congruent? 35 Which pair of shapes is similar but not congruent? A B Answer: B C D

Which of the following terms best describes the pair of figures? 36 Which of the following terms best describes the pair of figures? A congruent B similar C neither congruent nor similar Answer: B

Which of the following terms best describes the pair of figures? 37 Which of the following terms best describes the pair of figures? A congruent B similar C neither congruent nor similar Answer: A

Which of the following terms best describes the pair of figures? 38 Which of the following terms best describes the pair of figures? A congruent B similar C neither congruent nor similar Answer: C

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 red, the image is blue. Teacher Instructions: Congruent Translate 9 right Rotate 90 degree clockwise about (1,2) (x,y)  (x+9,y)  (y2 ,-x2 ) Congruent Translate 9 right Rotate 90 degree clockwise about (1, 2) (x, y) (x + 9, y) (y2, -x2)

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 red, the image is blue. Teacher Instructions: Congruent Reflect across x-axis Translate 2 left (x,y)  (x,-y)  (x2 - 2,y2 )

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 red, the image is blue. Teacher Instructions: Similar Dilate Scale Factor of 0.5 Reflect across y-axis (x, y) (0.5x, 0.5y) (-x2, y2)

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 red, the image is blue. Teacher Instructions: Similar Dilate Scale Factor of 2 Translate 4 up, 3 left (x, y) (2x, 2y) (x2-3, y2+4)

Special Pairs of Angles Return to Table Of Contents

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.

Vertical Angles are two angles that are opposite each other when two lines intersect. b c d In this example, the vertical angles are: Vertical angles have the same measurement. So:

Using what you know about complementary, supplementary and vertical angles, find the measure of the missing angles. b c a By Vertical Angles: By Supplementary Angles: Click Click

Are angles 2 and 4 vertical angles? 39 Are angles 2 and 4 vertical angles? Yes 1 2 3 4 No Answer: Yes

Are angles 2 and 3 vertical angles? 40 Are angles 2 and 3 vertical angles? Yes 1 2 3 4 No Answer: No

If angle 1 is 60 degrees, what is the measure of 41 If angle 1 is 60 degrees, what is the measure of angle 3? You must be able to explain why. A 30o B 60o 2 1 3 4 C 120o Answer: B – vertical angles are congruent D 15o

If angle 1 is 60 degrees, what is the measure of 42 If angle 1 is 60 degrees, what is the measure of angle 2? You must be able to explain why. A 30o B 60o C 120o Answer: C – They are supplemental angles. Supplemental angles add up to 180°. D 15o 2 1 3 4

ABC is adjacent to CBD How do you know? Adjacent Angles are two angles that are next to each other and have a common ray between them. This means that they are on the same plane and they share no internal points. A B C D ABC is adjacent to CBD How do you know? They have a common side (ray ) They have a common vertex (point B) 23 35

Adjacent or Not Adjacent? You Decide! b a b a b click to reveal click to reveal click to reveal Adjacent Not Adjacent Not Adjacent

Which two angles are adjacent to each other? 43 Which two angles are adjacent to each other? A 1 and 4 B 2 and 4 1 4 5 6 2 Answer: A 3

Which two angles are adjacent to each other? 44 Which two angles are adjacent to each other? A 3 and 6 B 5 and 4 2 1 Answer: A 5 4 3 6

A transversal is a line that cuts across two or more (usually parallel) lines. Q R B E F Interactive Activity-Click Here

In this diagram the corresponding angles are: Corresponding Angles are on the same side of the transversal and on the same side of the given lines. a b c d e f g h Transversal In this diagram the corresponding angles are: Click

Which are pairs of corresponding angles? 45 Which are pairs of corresponding angles? A 2 and 6 B 3 and 7 C 1 and 8 1 2 3 4 5 6 Answer: A & B 7 8

Which are pairs of corresponding angles? 46 Which are pairs of corresponding angles? A 2 and 6 B 3 and 7 6 3 C 1 and 8 2 5 4 1 Answer: B 8 7

Which are pairs of corresponding angles? 47 Which are pairs of corresponding angles? A 1 and 5 2 1 B 2 and 8 3 4 C 4 and 8 6 5 7 8 Answer: A, C

Which are pairs of corresponding angles 48 Which are pairs of corresponding angles A 2 and 4 4 5 B 6 and 5 8 1 C 7 and 8 2 6 D 1 and 3 7 3 Answer: A, B, C, D

In this diagram the alternate exterior angles are: Alternate Exterior Angles are on opposite sides of the transversal and on the outside of the given lines. a b c d e f g h l In this diagram the alternate exterior angles are: m Click n Which line is the transversal?

In this diagram the alternate interior angles are: Alternate Interior Angles are on opposite sides of the transversal and on the inside of the given lines. a b c d e f g h l In this diagram the alternate interior angles are: m Click n

In this diagram the same side interior angles are: Same Side Interior Angles are on same sides of the transversal and on the inside of the given lines. a b c d e f g h l In this diagram the same side interior angles are: m Click n

Are angles 2 and 7 alternate exterior angles? 49 Are angles 2 and 7 alternate exterior angles? Yes 1 3 5 7 2 4 6 8 m n l No Answer: No

Are angles 3 and 6 alternate exterior angles? 50 Are angles 3 and 6 alternate exterior angles? Yes 1 3 5 7 2 4 6 8 m n l No Answer: Yes

Are angles 7 and 4 alternate exterior angles? 51 Are angles 7 and 4 alternate exterior angles? Yes 1 3 5 7 2 4 6 8 m n l No Answer: No

Which angle corresponds to angle 5? 52 Which angle corresponds to angle 5? A 3 B 1 3 5 7 2 4 6 8 m n l 4 C 2 D 6 Answer: D

Which pair of angles are same side interior? 53 Which pair of angles are same side interior? A 3, 4 B 4, 7 1 3 5 7 2 4 6 8 m n l C 2, 4 D 6, 1 Answer: B

What type of angles are and ? 3 6 54 What type of angles are and ? 3 6 A Alternate Interior Angles B Alternate Exterior Angles C Corresponding Angles D Vertical Angles 1 3 5 7 2 4 6 8 m n l E Same Side Interior Answer: B

What type of angles are and ? 5 2 55 What type of angles are and ? 5 2 A Alternate Interior Angles B Alternate Exterior Angles C Corresponding Angles 1 3 5 7 2 4 6 8 m n l D Vertical Angles E Same Side Interior Answer: E

What type of angles are and ? 5 6 56 What type of angles are and ? 5 6 A Alternate Interior Angles B Alternate Exterior Angles C Corresponding Angles D Vertical Angles E Same Side Interior 1 3 5 7 2 4 6 8 m n l Answer: C

Are angles 5 and 2 alternate interior angles? 57 Are angles 5 and 2 alternate interior angles? Yes 1 3 5 7 2 4 6 8 m n l No Answer: No

Are angles 5 and 7 alternate interior angles? 58 Are angles 5 and 7 alternate interior angles? Yes 1 3 5 7 2 4 6 8 m n l No Answer: No

Are angles 7 and 2 alternate interior angles? 59 Are angles 7 and 2 alternate interior angles? Yes 1 3 5 7 2 4 6 8 m n l No Answer: Yes

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

If parallel lines are cut by a transversal then: Special Case!!! 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 1 3 5 7 2 4 6 8 n m click l These Special Cases can further be explained using the transformations of reflections and translations

Which angles are congruent to the given angle? 61 Given the measure of one angle, find the measures of as many angles as possible. Which angles are congruent to the given angle? A <4, <5, <6 4 5 6 2 7 1 8 l m n B <5, <7, <1 C <2 Answer: B D <5, <1

What are the measures of angles 4, 6, 2 and 8? 62 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 4 5 6 2 7 1 8 l m n C 130o Answer: C

Which angles are congruent to the given angle? 63 Given the measure of one angle, find the measures of as many angles as possible. Which angles are congruent to the given angle? l A <4 m B <4, <5, <3 3 1 C <2 5 7 Answer: A D <8 2 4 n 8

What are the measures of angles 2, 4 and 8 respectively? 64 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, 550 B 35o, 35o, 35o C 145o, 35o, 145o l Answer: C m 3 1 5 7 2 4 n 8

Answer: A

Answer: C

Answer: D

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

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

Let's draw a line through B parallel to AC. We then have a two parallel lines cut by a transversal. Number the angles and use what you know to prove the sum of the measures of the angles equals 180. l m n p B A C 1 2

l m n p B A C 1 2 since if 2 parallel lines are cut by a transversal, then alternate interior angles are congruent. is supplementary with since if 2 parallel lines are cut by a transversal, then same side interior angles are supplementary. 3. Therefore,

Let's look at this another way... m n p B A C 2 1 and since if 2 parallel lines are cut by a transversal, then alternate interior angles are congruent. Since all three angles form a straight line, the sum of the angles is 180.

Let's prove the Exterior Angle Theorem - The measure of the exterior angle of a triangle is equal to the sum of the remote interior angles. ABC B 1 A C Exterior Angle Remote Interior Angles

Let's draw a line through B parallel to AC. We then have a two parallel lines cut by a transversal. Number the angles and use what you know to prove the measure of angle 1 = the sum of the measures of angles B and C. l m n p B A C 2 1

l m n p B A C 1 3 2 since if 2 parallel lines are cut by a transversal, then alternate interior angles are congruent. 2. since if 2 parallel lines are cut by a transversal, then alternate interior angles are congruent. 4. Therefore,

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Answer: 32 degrees

Answer: C

Answer: 110 degrees