UNIT 1 SIMILARITY, CONGRUENCE, AND PROOFS Lesson 4: Exploring Congruence Instruction
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1 Prerequisite Skills This lesson requires the use of the following skills: constructing perpendicular bisectors copying a segment copying an angle Introduction Think about trying to move a drop of water across a flat surface. If you try to push the water droplet, it will smear, stretch, and transfer onto your finger. The water droplet, a liquid, is not rigid. Now think about moving a block of wood across the same flat surface. block of wood is solid or rigid, meaning it maintains its shape and size when you move it. You can push the block and it will keep the same size and shape as it moves. In this lesson, we will eamine rigid motions, which are transformations done to an object that maintain the object s shape and size or its segment lengths and angle measures. Key oncepts Rigid motions are transformations that don t affect an object s shape and size. This means that corresponding sides and corresponding angle measures are preserved. When angle measures and sides are preserved they are congruent, which means they have the same shape and size. The congruency symbol ( ) is used to show that two figures are congruent. The figure before the transformation is called the preimage. The figure after the transformation is the image. orresponding sides are the sides of two figures that lie in the same position relative to the figure. In transformations, the corresponding sides are the preimage and image sides, so and are corresponding sides and so on. orresponding angles are the angles of two figures that lie in the same position relative to the figure. In transformations, the corresponding vertices are the preimage and image vertices, so and are corresponding vertices and so on. Transformations that are rigid motions are translations, reflections, and rotations. Transformations that are not rigid motions are dilations, vertical stretches or compressions, and horizontal stretches or compressions. U
2 Translations translation is sometimes called a slide. In a translation, the figure is moved horizontally and/or vertically. The orientation of the figure remains the same. onnecting the corresponding vertices of the preimage and image will result in a set of parallel lines. Translating a Figure Given the Horizontal and Vertical Shift. Place your pencil on a verte and count over horizontally the number of units the figure is to be translated.. Without lifting your pencil, count vertically the number of units the figure is to be translated.. Mark the image verte on the coordinate plane.. Repeat this process for all vertices of the figure.. onnect the image vertices. Reflections reflection creates a mirror image of the original figure over a reflection line. reflection line can pass through the figure, be on the figure, or be outside the figure. Reflections are sometimes called flips. The orientation of the figure is changed in a reflection. In a reflection, the corresponding vertices of the preimage and image are equidistant from the line of reflection, meaning the distance from each verte to the line of reflection is the same. The line of reflection is the perpendicular bisector of the segments that connect the corresponding vertices of the preimage and the image. U
3 Reflecting a Figure over a Given Reflection Line. Draw the reflection line on the same coordinate plane as the figure.. If the reflection line is vertical, count the number of horizontal units one verte is from the line and count the same number of units on the opposite side of the line. Place the image verte there. Repeat this process for all vertices.. If the reflection line is horizontal, count the number of vertical units one verte is from the line and count the same number of units on the opposite side of the line. Place the image verte there. Repeat this process for all vertices.. If the reflection line is diagonal, draw lines from each verte that are perpendicular to the reflection line etending beyond the line of reflection. opy each segment from the verte to the line of reflection onto the perpendicular line on the other side of the reflection line and mark the image vertices.. onnect the image vertices. Rotations rotation moves all points of a figure along a circular arc about a point. Rotations are sometimes called turns. In a rotation, the orientation is changed. The point of rotation can lie on, inside, or outside the figure, and is the fied location that the object is turned around. The angle of rotation is the measure of the angle created by the preimage verte to the point of rotation to the image verte. ll of these angles are congruent when a figure is rotated. Rotating a figure clockwise moves the figure in a circular arc about the point of rotation in the same direction that the hands move on a clock. Rotating a figure counterclockwise moves the figure in a circular arc about the point of rotation in the opposite direction that the hands move on a clock. U-0
4 Rotating a Figure Given a Point and ngle of Rotation. Draw a line from one verte to the point of rotation.. Measure the angle of rotation using a protractor.. Draw a ray from the point of rotation etending outward that creates the angle of rotation.. opy the segment connecting the point of rotation to the verte (created in step ) onto the ray created in step.. Mark the endpoint of the copied segment that is not the point of rotation with the letter of the corresponding verte, followed by a prime mark ( ). This is the first verte of the rotated figure.. Repeat the process for each verte of the figure.. onnect the vertices that have prime marks. This is the rotated figure. ommon Errors/Misconceptions creating the angle of rotation in a clockwise direction instead of a counterclockwise direction and vice versa reflecting a figure about a line other than the one given mistaking a rotation for a reflection misidentifying a translation as a reflection or a rotation U
5 Guided Practice.. Eample Describe the transformation that has taken place in the diagram below. y ' '. Eamine the orientation of the figures to determine if the orientation has changed or stayed the same. Look at the sides of the triangle. Side length Preimage orientation Image orientation Shortest Longest Intermediate ottom of triangle and horizontal Right side of triangle going from top left to bottom right Left side of triangle and vertical ottom of triangle and horizontal Right side of triangle going from top left to bottom right Left side of triangle and vertical The orientation of the triangles has remained the same. U-
6 . onnect the corresponding vertices with lines. y ' ' The lines connecting the corresponding vertices appear to be parallel.. nalyze the change in position. heck the horizontal distance of verte. To go from to horizontally, the verte was shifted to the right units. Vertically, verte was shifted down units. heck the remaining two vertices. Each verte slid units to the right and units down. 0 y ' ' U-
7 Eample Describe the transformation that has taken place in the diagram below. D' E' y E D F' ' ' F. Eamine the orientation of the figures to determine if the orientation has changed or stayed the same. Look at the sides of the figures and pick a reference point. good reference point is the outer right angle of the figure. From this point, eamine the position of the arms of the figure. rm Preimage orientation Image orientation Shorter Pointing upward from the corner of the figure with a negative slope at the end of the arm Pointing downward from the corner of the figure with a positive slope at the end of the arm Longer Pointing to the left from the corner of the figure with a positive slope at the end of the arm Pointing to the left from the corner of the figure with a negative slope at the end of the arm (continued) U-
8 The orientation of the figures has changed. In the preimage, the outer right angle is in the bottom right-hand corner of the figure, with the shorter arm etending upward. In the image, the outer right angle is on the top righthand side of the figure, with the shorter arm etending down. lso, compare the slopes of the segments at the end of the longer arm. The slope of the segment at the end of the arm is positive in the preimage, but in the image the slope of the corresponding arm is negative. similar reversal has occurred with the segment at the end of the shorter arm. In the preimage, the segment at the end of the shorter arm is negative, while in the image the slope is positive.. Determine the transformation that has taken place. Since the orientation has changed, the transformation is either a reflection or a rotation. Since the orientation of the image is the mirror image of the preimage, the transformation is a reflection. The figure has been flipped over a line.. Determine the line of reflection. onnect some of the corresponding vertices of the figure. hoose one of the segments you created and construct the perpendicular bisector of the segment. Verify that this is the perpendicular bisector for all segments joining the corresponding vertices. This is the line of reflection. The line of reflection for this figure is y =. 0 y y = E - F - D - - D' E' F' ' ' U-
9 Eample Describe the transformation that has taken place in the diagram below. y R ' '. Eamine the orientation of the figures to determine if the orientation has changed or stayed the same. Look at the sides of the triangle. Side length Preimage orientation Image orientation Shortest Right side of triangle and Top of triangle and vertical horizontal Longest Intermediate Diagonal from bottom left to top right of triangle ottom side of triangle and horizontal The orientation of the triangles has changed. Diagonal from top left to bottom right of triangle Right side of triangle and vertical U-
10 . Determine the transformation that has taken place. Since the orientation has changed, the transformation is either a reflection or a rotation. Since the orientation of the image is NOT the mirror image of the preimage, the transformation is a rotation. The figure has been turned about a point. ll angles that are made up of the preimage verte to the reflection point to the corresponding image verte are congruent. This means that R R R. y R ' ' U-
11 Eample Rotate the given figure º counterclockwise about the origin. 0 y reate the angle of rotation for the first verte. onnect verte and the origin with a line segment. Label the point of reflection R. Then, use a protractor to measure a angle. Use the segment from verte to the point of rotation R as one side of the angle. Mark a point X at. Draw a ray etending out from point R, connecting R and X. opy R onto RX. Label the endpoint that leads away from the origin. 0 y R X ' U-
12 . reate the angle of rotation for the second verte. onnect verte and the origin with a line segment. The point of reflection is still R. Then, use a protractor to measure a angle. Use the segment from verte to the point of rotation R as one side of the angle. Mark a point Y at. Draw a ray etending out from point R, connecting R and Y. opy R onto RY. Label the endpoint that leads away from the origin. y 0 R ' Y U-
13 . reate the angle of rotation for the third verte. onnect verte and the origin with a line segment. The point of reflection is still R. Then, use a protractor to measure a angle. Use the segment from verte to the point of rotation R as one side of the angle. Mark a point Z at. Draw a ray etending out from point R, connecting R and Z and continuing outward from Z. opy R onto RZ. Label the endpoint that leads away from the origin. 0 y R ' ' Z. onnect the rotated points. The connected points,, and form the rotated figure. 0 y ' ' U-0
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Geometry. Geometry. Domain Cluster Standard. Congruence (G CO)
Domain Cluster Standard 1. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance
Pre-Algebra Notes Unit 13: Angle Relationships and Transformations
Pre-Algebra Notes Unit 13: Angle Relationships and Transformations Angle Relationships Sllabus Objectives: (7.1) The student will identif measures of complementar, supplementar, and vertical angles. (7.2)
Learning Log Title: CHAPTER 6: TRANSFORMATIONS AND SIMILARITY. Date: Lesson: Chapter 6: Transformations and Similarity
Chapter 6: Transformations and Similarity CHAPTER 6: TRANSFORMATIONS AND SIMILARITY Date: Lesson: Learning Log Title: Date: Lesson: Learning Log Title: Chapter 6: Transformations and Similarity Date: Lesson:
Focus of this Unit: Connections to Subsequent Learning: Approximate Time Frame: 4-6 weeks Connections to Previous Learning:
Approximate Time Frame: 4-6 weeks Connections to Previous Learning: In Grade 8, students are introduced to the concepts of congruence and similarity through the use of physical models and dynamic geometry
Pearson Mathematics Geometry Common Core 2015
A Correlation of Pearson Mathematics Geometry Common Core 2015 to the Common Core State Standards for Bid Category 13-040-10 A Correlation of Pearson, Common Core Pearson Geometry Congruence G-CO Experiment
Rotations. Essential Question How can you rotate a figure in a coordinate plane?
11.3 Rotations Essential Question How can ou rotate a figure in a coordinate plane? Rotating a Triangle in a oordinate lane ONSTRUTING VILE RGUMENTS To be proficient in math, ou need to use previousl established
Test #1: Chapters 1, 2, 3 Test #2: Chapters 4, 7, 9 Test #3: Chapters 5, 6, 8 Test #4: Chapters 10, 11, 12
Progress Assessments When the standards in each grouping are taught completely the students should take the assessment. Each assessment should be given within 3 days of completing the assigned chapters.
Russell County Pacing Guide
August Experiment with transformations in the plane. 1. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment based on the undefined notions of point, line, distance
Students are not expected to work formally with properties of dilations until high school.
Domain: Geometry (G) Cluster: Understand congruence and similarity using physical models, transparencies, or geometry software. Standard: 8.G.1. Verify experimentally the properties of rotations, reflections,
Transformations Geometry
Transformations Geometry Preimage the original figure in the transformation of a figure in a plane. Image the new figure that results from the transformation of a figure in a plane. Example: If function
Lesson 9.1 Properties of Transformations
Lesson 9.1 roperties of Transformations Name eriod Date In Eercises 1 3, draw the image according to the rule and identif the tpe of transformation. 1. (, ) (, ) 2. (, ) ( 4, 6) 3. (, ) (4, ) 6 4 2 6 4
1-8 Exploring Transformations
1-8 Exploring Transformations Warm Up Lesson Presentation Lesson Quiz 2 Warm Up Plot each point. D 1. A(0,0) 2. B(5,0) 3. C( 5,0) 4. D(0,5) 5. E(0, 5) 6. F( 5, 5) C A F E B Objectives Apply transformations
Homework for Section 5.1
Homework for Section 5.1 1. reate the rotation R(T) 2. reate the reflection F(T) of the triangle T shown below 90 degrees of the triangle T shown below across clockwise about the center point of rotation.
Set the Sails! Purpose: Overview. TExES Mathematics 4-8 Competencies. TEKS Mathematics Objectives.
Set the Sails! Purpose: Participants will use graphing technology to investigate reflections, translations, rotations, and sequences of reflections and translations in the coordinate plane. They will give
What You ll Learn. Why It s Important
Many artists use geometric concepts in their work. Think about what you have learned in geometry. How do these examples of First Nations art and architecture show geometry ideas? What You ll Learn Identify
Unit 1 Unit 1 A M. M.Sigley, Baker MS. Unit 1 Unit 1. 3 M.Sigley, Baker MS
A M S 1 2 G O E A B 3 4 LINE POINT Undefined No thickness Extends infinitely in two directions Designated with two points Named with two capital letters or Undefined No size Named with a capital letter