8th Grade Math UNIT 1 Transformations, Congruence, and Similarity Unit Essential Questions   What are the transformations? How do you construct and use angle relationships? How can transformations and angle relationships be used to depict real-world situations?

Transformations MGSE8.G.3: Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. MGSE8.G.4: Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations: given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.

Warm Up/Vocabulary A coordinate plane is formed by the intersection of two number lines. x-axis - the horizontal axis y-axis - the vertical axis The origin is the point where the x and y axes intersect. Ordered pair - names the location of the point in the plane, usually written (x, y)

Dilations

EXAMPLE 1: Writing the Rule If the scale factor is 3, how would you write the rule?. (x,y)  (3x,3y)

EXAMPLE 2: Find the Coordinates Triangle ABC has vertices A(0,2), B(4,4), and C(-1,4). What are the vertices of its prime image it the scale factor is 4? c) (0,8) (16,16) (-4,16)

EXAMPLE 3: Enlarging an Object Use the coordinates to draw the original figure (pre-image) and its copied figure (prime) if the scale factor is 2. . (1,1) (3,-4) c) (-4,5) (2,2) (6,-8) (-8,10)

EXAMPLE 4: Reducing an Object Use the coordinates to draw the original figure (pre-image) and its copied figure (prime) if the scale factor is 1/3. (9,3) (0,-6) c) (-3,6) (3,1) (0,-2) (-1,2)

Translations

EXAMPLE 1: Writing the Rule If A is 19 and B is 89, how would you write the rule? (x,y)  (x+19,y+89) B. If A is -3 and B is 28, how would you write the rule? (x,y)  (x-3,y+28)

EXAMPLE 2: Find the Coordinates Triangle ABC has vertices A(0,2), B(4,4), and C(-1,4). What are the vertices of its prime image it the rule is (x-2,y+6)? A: (-2,8) B: (2,10) C: (-3,10) B. Triangle HIJ has vertices H(8,-16), I(19,89), and J(-5,101). What are the vertices of its prime image it the rule is (x-34,y-15)? A: (-26,-31) B: (-15,73) C: (-39,85)

EXAMPLE 3: Translating an Object Use the coordinates to draw the original figure (pre-image) and its copied figure (prime). Rule: (x+4,y-3) (1,1) (3,-4) c) (-4,5) (5,-2) (7,-7) (0,2)

EXAMPLE 3: Translating an Object Use the coordinates to draw the original figure (pre-image) and its copied figure (prime). Rule: (x-5,y) (9,3) (0,-6) c) (-3,6) (4,3) (-5,-6) (-8,6)

Reflections

EXAMPLE 1: Find the Coordinates Triangle ABC has vertices A(0,2), B(4,4), and C(-1,4). What are the vertices of its prime image if the rule is reflect across the x-axis? A: (0,-2) B: (4,-4) C: (-1,-4) B. Triangle HIJ has vertices H(8,-16), I(19,89), and J(-5,101). What are the vertices of its prime image if the rule is reflect across the y-axis? A: (-8,-16) B: (-19,89) C: (5,101)

EXAMPLE 2: Reflecting an Object Use the coordinates to draw the original figure (pre-image) and its copied figure (prime). Rule: reflect across x = 1 (1,1) (3,-4) c) (-4,5) (1,1) (-1,-4) (6,5)

EXAMPLE 2: Reflecting an Object Use the coordinates to draw the original figure (pre-image) and its copied figure (prime). Rule: reflect across y = 2 (9,3) (0,-6) c) (-3,6) (9,1) (0,10) (-3,-2)

Rotations

EXAMPLE 1: Find the Coordinates Triangle ABC has vertices A(0,2), B(4,4), and C(-1,4). What are the vertices of its prime image it is rotation 90° CCW about the origin? A: (-2,0) B: (-4,4) C: (-4,-1) B. Triangle HIJ has vertices h(8,-16), i(19,89), and j(-5,101). What are the vertices of its prime image it is rotated 270° CW about the origin A: (-8,-16) B: (-19,89) C: (5,101)

EXAMPLE 2: Rotating an Object Use the coordinates to draw the original figure (pre-image) and its copied figure (prime). Rule: 180º rotations about the origin (1,1) (3,-4) c) (-4,5) (-1,-1) (-3,4) (4,-5)

EXAMPLE 2: Rotating an Object Use the coordinates to draw the original figure (pre-image) and its copied figure (prime). Rule: 90º CW about the origin (9,3) (0,-6) c) (-3,6) (3,-9) (-6,0) (6,3)

RATE YOUR UNDERSTANDING Transformations MGSE8.G.3: Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. MGSE8.G.4: Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations: given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them. RATING LEARNING SCALE 4 I am able to • Represent a transformations and apply them real-world situations 3 • Construct and transform a figure from given points. • Interpret graphs and conclude which transformation, or combination of, was/were performed. 2 • Identify each of the transformations. 1 • Understand how to read and plot points on a coordinate plane.   TARGET

Angle Relationships MGSE8.G.1: Verify experimentally the congruence properties of rotations, reflections, and translations: lines are taken to lines and line segments to line segments of the same length; angles are taken to angles of the same measure; parallel lines are taken to parallel lines. MGSE8.G.2: Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them. MGSE8.G.5: Use informal argument to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles.

Warm Up/Vocabulary E A and D D A C B A 1. Acute Angle ______ A. 2. Adjacent Angles ___________ B. 3. Complementary Angles ______ 4. Linear Pair ______ C. 5. Obtuse Angle ______ D. Right Angle ______ 7. Supplementary Angles ______ E. A and D D A C B A

Naming Angles

EXAMPLE 1: Find the Missing Piece Name the type(s) of angle(s) and solve for the unknown angle. a) Supplementary Linear Pair Adjacent 128º Complimentary Adjacent 28º

Finding Missing Angles

EXAMPLE 1: Find the Missing Piece Solve for the unknown variable and the size of each angle. a) x = 29 Angles from left to right = 32º, 28º, and 30º. x = 19 Angels from left to right = 99º, 17º, and 64º.

Angle Relationships

Warm Up/Vocabulary 1&7 and 2&8 1&8 and 2&7 3&6 and 4&5 1. Alternate Exterior Angles _____________ 2. Alternate Interior Angles _____________ 3. Corresponding Angles _______ 4. Same Side Exterior Angles _______ 5. Same Side Interior Angles _______ 6. Vertical Angles _______ 3&6 and 4&5 4&6 and 3&5 1&5, 2&6, 3&7, and 4&8 1&3, 2&4, 5&7, and 6&8 Based on the definitions you’ve found, identify an appropriate angle relationship.

Congruent Relationships Supplementary Relationships   Alternate Exterior Angles Same Side Exterior Angles Alternate Interior Angles Same Side Interior Angles Corresponding Angles Vertical Angles

EXAMPLE 1: Identify the Appropriate Relationship Give the correct relationship between the given angles ∠1 and ∠7 ∠5 and ∠7 C. ∠3 and ∠7 Alternate Exterior Vertical Corresponding 

EXAMPLE 2: Identify the Measure of the Missing Angle Using what you know about angle relationships, find the measure of the missing angles. ∠1: ________ ∠6: ________ ∠2: ________ ∠5: ________ 135º 45º 45º  45º

EXAMPLE 3: Find the Missing Value Find the value of the variable indicated. Remember to check your work. X = ____ B. Y = _____ 20 27

RATE YOUR UNDERSTANDING Angle Relationships MGSE8.G.1: Verify experimentally the congruence properties of rotations, reflections, and translations: lines are taken to lines and line segments to line segments of the same length; angles are taken to angles of the same measure; parallel lines are taken to parallel lines. MGSE8.G.2: Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them. MGSE8.G.5: Use informal argument to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. RATING LEARNING SCALE 4 I am able to • Use angle relationships to solve real-world problems. 3 • Construct angle relationships. • Interpret graphs and conclude which transformation, or combination of, was/were performed. 2 • Identify angle relationships. 1 • Understand the definitions of the different angle relationships.   TARGET