Objectives Line Segment Addition Distance formula and midpoint

Objectives Line Segment Addition, Distance formula and midpoint Vocabulary Point, line, ray, plane, line segment Assignments Honors: 5, 9, 12, 13, 15 -20, 22, 24, 28, 29 Regular: 5, 9, 13, 17 -20, 22, 23, Both: 3

Day 2 BASIC VOCABULARY Notation is VERY important 4

Explain 1: Postulates A postulate is a statement that is accepted as true without proof 5

What are Constructions? In this lesson we will use these to “build” -Copying an Angle -Angle Bisector You will see more of these throughout Geometry Construction? Use link on teacher website. http: //www. mathopenref. com/construction. html 6

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Explain 2: Distance Between Points Always Sketch 1. Subtracts x’s square this # 2. Subtract y’s square this # 3. Add steps 1 & 2 Square ROOT this # Alternative for Distance Formula: _____________ 8

Explain 2: Distance Between Points 10

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Explain 3: Finding Midpoint 13

Explain 4: Midpoint Formula 14

-3 -1 0 2 So, M (lies / doesn’t lie) on the y-axis since ___________ 15

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Objectives Name Angles, Angle Bisector, Constructions Vocabulary Angle, Vertex, Side, Angle Bisector Assignments Honors: Evaluate 1. 2: 5, 6, 7, 9, 12, 15, 16, 20, 21, 22 Evaluate 1. 4: 5, 6, 8, 11, 17, 19, 23, 24, 26, 28 Regular: Evaluate 1. 2 4, 5, 6, 7, 8, 12, 15, 20 Evaluate 1. 4: 5, 6, 11, 17, 19, 23, 26 19

Explore: Copying an Angle Construction? Watch video on teacher website. 20

Explain 1: Naming Angles or Important Notes: -Order Matters -The middle letter is the vertex 21

5. Name ALL the angles, in as many ways possible 22

Explain 2: Measuring Angles • 23

This Photo by Unknown Author is licensed under CC BY-SA 24

Explain 3: Angle Addition Postula 25

Explain 3. 5: Angle Bisectors Example 1: A. Find the value of x B. Find the measure of the entire angle A. Finding x: B. Finding angle: 26

Angle Bisectors Example 2 There are SEVERAL ways to solve. Here is one way (by doubling the angle)= 27

Construction? Watch video on teacher website. This Photo by Unknown Author is licensed under CC BY-SA 28

Objectives Use Theorems and properties to prove a statement Vocabulary Theorem, counter example, Conditional Statement Assignments Honors: Evaluate 1. 2: 5, 6, 7, 9, 12, 15, 16, 20, 21, 22 Evaluate 1. 4: 5, 6, 8, 17, 19, 23, 24, 26, 28 Regular: Evaluate 1. 2 4, 5, 6, 7, 8, 12, 15, 20 Evaluate 1. 4: 5, 6, 11, 17, 19, 23, 26 30

Basic Proof Vocabulary 31

Properties of Equalities From now on, use these Properties to justify steps in basic and complex math problems • Instead of saying “add 5 on both sides because what we do to one side we do to the other” say “add 5 on both sides because of the Addition Property of Equality” 32

Step Reason Given is what we say at the beginning with the question given 33

Step Reason 34

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• Explain 2: Using Postulates about Segments and Angles 36

We did this problem in section 1. 2 Statement Reason 37

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Explain 3: Using Postulates about Lines and Planes (Part 1) 40

Explain 3: Using Postulates about Lines and Planes (Part 2) 41

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Objectives Use transformations using rules Vocabulary Transformation, preimage, image Assignments Honors: 1, 3, 5, 10, 15, 16, 17 Regular: 2, 4, 5, 10, 15, 45

Basic Transformation Vocabulary • A transformation is a function that changes the position, shape, and/or size od a figure • The input of a transformation od the preimage, like A • The output is the image, like A’ (A prime) • Translation, reflections and rotations are three types of transformation. 46

Explain 1: Rigid vs Nonrigid • Some transformations preserve length and angle measure, others do not • A rigid motion changed position without affecting size or shape of the figure. • Examples: translation, reflections, and rotations Steps: 1) Look for patters in the coordinates 2) Compare lengths of the preimage and image of corresponding segments 3) Compare angles of corresponding angles 47

Types of Transformation Decide if they are Rigid or NON-rigid -Up -Down -Left -Right We will discuss this later This Photo by Unknown Author is licensed under CC BY-NC-SA 48

Transformation Example 49

Recall Finding Distance 50

PREIMAGE Distance P to Q Distance P’ to Q’ Distance P to R Distance P’ to R’ Distance Q to R Distance Q’ to R’ Recall Finding Distance 51

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Explain 2: Non Rigid Motions Steps: 1) Look for patterns in the coordinates (usually multiplication) 2) Compare lengths of the preimage and image of corresponding segments 53

The y-value is being multiplied by ___ Because this CHANGES sides, this is a _______ transformation. 54

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Translations, Rotations, Reflections: The Basics Objectives Identify transformation, and its characteristics Vocabulary Transformation, rotational symmetry, line symmetry Assignments Honors: (Evaluate 2. 4) 2, 3, 4, 6, 7, 8, 10, 13, 16 Regular: (Evaluate 2. 4) 2, 4, 7, 8, 14, 16 Both: Copy Do Now #5 Notes from website in composition notebook 57

In geometry, a transformation is a way to change the position of a figure. 59

TRANSLATION A translation is a transformation that slides a figure across a plane or through space. With translation all points of a figure move the same distance and the same direction. 62

TRANSLATION Basically, translation means that a figure has moved. An easy way to remember what translation means is to remember… A TRANSLATION IS A SLIDE A translation is usually specified by a direction and a distance, AKA a vector. 63

REFLECTION A reflection is a transformation that flips a figure across a line. A REFLECTION IS FLIPPED OVER A LINE. 68

REFLECTION Sometimes, a figure has reflectional symmetry. This means that it can be folded along a line of reflection within itself so that the two halves of the figure match exactly, point by point. Basically, if you can fold a shape in half and it matches up exactly, it has reflectional symmetry. 72

Your Turn How many lines of symmetry does each shape have? Do you see a pattern? 76

ROTATION A rotation is a transformation that turns a figure about (around) a point or a line. Basically, rotation means to spin a shape. The point a figure turns around is called the center of rotation. The center of rotation can be on or outside the shape. 79

ROTATION What does a rotation look like? center of rotation A ROTATION MEANS TO TURN A FIGURE 80

ROTATION When some shapes are rotated they create a special situation called rotational symmetry. to spin a shape the exact same 87

ROTATIONAL SYMMETRY A shape has rotational symmetry if, after you rotate less than one full turn, it is the same as the original shape. As this shape is rotated 360 , is it ever the same before the shape returns to its original direction? Yes, when it is rotated 90 it is the same as it was in the beginning. So this shape is said to have rotational symmetry. The angles of rotation are: 90 +90 Divide by how many times the shape is the same after a full rotation then add that until you reach 360. 88

CONCLUSION We just discussed three types of transformations. See if you can match the action with the appropriate transformation. FLIP SLIDE TURN REFLECTION TRANSLATION ROTATION 92

Putting Lines of Symmetry and Rotational Symmetries Together. ONE EXAMPLE 4 Your Turns KAHo. OOOoo. T!! 93

Example: 94

Honors Practice 95

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Translations, Rotations, Reflections: Part 2 Objectives Connect the rule to the transformation Vocabulary Vector Honors & Regular: Evaluate 2. 1: 10, 12 -14 Evaluate 2. 2: 9, 11, 13, 14 Evaluate 2. 3: 10 -12, 15 Both: Assignments 100

Translations as Vectors • How could vectors be connected to translations? 101

Explain 1: Translations 102

Explain 2: Translation in Coordinate Plane Vector <5, 3> This is “a” Left/right This is “b” Up/down Getting NEW Point from OLD point: 103

Verbal Description: Function Description: Step 1: Copy Vector on every vertex Step 2: Draw Segments 104

Verbal Description: Function Description: 105

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Honors: Both Regular: #5 109

Explain 3: Identifying the Vector 110

When will your Reflection show who you are? 111

Explain 1: Reflect using Perpendicular Bisectors • Points of Preimage and reflected image must me equidistant from the line of reflection. • This distance bisects the line of reflection 112

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Explain 2: Reflections as Coordinate Points 115

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Explain 3: Find the Line of Reflection To find the line of reflection, look for the midpoint between corresponding points. Why does this work? 119

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Rotations 121

Explain 2: Rotations as Coordinate Points When not specified, assume Counterclockwise Click here for a TRICK to remember this 122

(About the Origin) 123

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Rotations NOT about the Origin S – Subtract point of rotation A – Apply Rule A – Add point of rotation G – Graph, if necessary 126

Video 127

Your Turn Counterclockwise; 270 degrees 128

Objectives Use a sequence of transformation Vocabulary sequence Assignments Honors: 2 -5, 8, 13 -19, 23, when there are vector, write out both verbal and coordinate point meaning. Regular: 2, 4, 5, 13 -19 Copy Do Now Notes #7 130

Explain 1: Combining Rigid Transformations 131

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Explain 2: Combining Non-rigid Get the points from the Graph A B C 134

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(as a class discussion) 136

Objectives Prove that figures are congruent using rigid motions Vocabulary … Assignments Honors: 2 -5, 7, 9, 10, 12, 14, 17, 28 Regular: 1, 4, 5, 7, 10, 13, 17 138

Explain 1: Determining Congruence 139

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Explain 2: Find Sequence of Rigid Motions Reflection over the Y-axis Translation Reflection (-x, y) (x + 2, y + 10) Translation 141

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3. 2 Classwork • Honors: 2 -5, 7, 9, 10, 12, 14, 17, 28 Regular: 1, 4, 5, 7, 10, 13, 17 • Copy Do Now Notes #8 onto composition notebook. LAST ONE. Woohoo. Reminders: q…. 143

Objectives Congruent Corresponding Parts prove congruent figures. Vocabulary CPCTC Assignments 3. 3 Honors: 2, 3, 7, 10, 15, 20, 21, 23 Discovering Geometry Page 252 -3: 7, 11, 25, 26, 29, 30 3. 3 Regular: 2, 3, 6, 7, 10, 21 3. 2 Honors: 2 -5, 7, 9, 10, 12, 14, 17, 28 3. 2 Regular: 1, 4, 5, 7, 10, 13, 17 144