Unit 6: Rigid Motion Congruency

Transcription

1 Name: Geometry Period Unit 6: Rigid Motion Congruency In this unit you must bring the following materials with you to class every day: Please note: Pencil This Booklet A device This booklet will be scored and checked on the last day of the unit using the rating scale shown below Some days you will have additional handouts to support your understanding of the learning goals in that lesson. Keep these in a folder and bring to class every day. All homework is in this booklet Answer keys will be posted as usual for each daily lesson on our website This is only 3.5 lessons so stay on top of your work and you will be prepared.

2 6-1 Notes Lesson 6-1:Rigid Motion Recap Today s Goal: What are concepts we remember about Rigid Motions? Re-activating our knowledge! Partner work directions: Examine the movements shown below. Answer the following questions for each movement, use the space provided. Use proper vocabulary and clear explanations! 1. Identify each type of rigid motion you see below: a) b) c) How come m didn t move? This is an example of a: This is an example of a: This is an example of a: 2. In each of the above diagrams, circle the pre-image. 3. Why are all of these transformations above rigid motions? What does that mean? Recall: Congruence If parts of triangles are, then the triangles are. Two triangles are congruent, then their parts are congruent.

3 Special Relationship: Think through this with a partner, we ll come back together and explain the relationship together! a. What type of transformation is shown right? b. Fill in: Each vertex in and its corresponding vertex in are to the line of reflection. Fill in: This means that the from A to line m and the from A to line m are. (Same for B & B and C & C ). c. Draw a segment connecting pre-image B and image B. Fill in: Segment and line m are to each other. Is this true when we connect all corresponding points on the figure? (Sketch them in)! d. Using your responses above, what is another name for line m? (Look at the line with respect to a segment formed by a pre-image and an image point.) Reflections The line of reflection between a pre-image and its reflected image is also the of the segment formed by corresponding points in an image and pre-image. Refining our language. Above you see a reflection over line m. We can DESCRIBE what specific point in the preimage will land on a specific point in the image. We call this mapping. After reflected over line m, A maps onto After reflected over line m maps on to C After reflected over line m maps onto Paying attention to detail What key components do we need include when discussing each of the following transformations? Translations Rotations Reflections

4 Practice 1. In the figure below, the triangle on the left can be mapped to the triangle on the right by a rotation about point P. Fill in the following blanks. After a rotation centered at P, A maps onto After a rotation, maps onto Y After a rotation centered at P, C Z 2. Describe the rigid motion you see by filling in the blanks: a) ABC can be mapped onto A B C by a along the BB b) ABC can be mapped onto A B C by a about point So that A A. 3. In the figure shown right, is a line of reflection. Determine whether the following statements are true or false. a) is the perpendicular bisector of segment b) c) J = J why?

5 4. Given in the figure below, line is the perpendicular bisector of and of. a. In terms of rigid motions, what transformation maps point A to point B? How do you know? b. Explain why using rigid motions. (Think! It is kind of like CPCTC!) In this unit, it s not just what you say, it s how you say it! When you finish this practice be sure to check the key so your teacher can help you make sure your wording is right!

6 6-1 HOMEWORK Answer each of the following questions. Don t forget to check your work online! Take this pre-assessment first to launch into this unit! Pre-Assessment: 1. Label the image and the pre-image in the following diagram shown right: Pre-Image: Image: 2. The three types of rigid motions are:,, 3. Describe what rigid motion you see taking place: a) b) c) b) Which of the following diagrams above has a vector? What does a vector show? What s the name of the vector? 4. a. What is the transformation shown below? Is it a rigid motion? Explain! b. Name a vector drawn in the diagram using vector notation.

7 5. A translation maps the figure on the left to the figure on the right. Solve for b and c. 6. Use the diagram below to answer a-c. a) What two transformations would have to be performed to map ABCD onto A B C D? Be specific! b) In the diagram above, (fill in the blanks)! maps onto A B B C maps onto c) Are the two figures congruent? Explain! 7. a) In the diagram below, which single transformation was used to map triangle A onto triangle B? 1) line reflection 2) rotation 3) dilation 4) translation b) Why is triangle A congruent to triangle B?

8 8. Three rigid motions are to be performed on square. first rigid motion is the reflection through line. The second rigid motion is a 90 clockwise rotation around the center of the square. Describe the third rigid motion that will ultimately map back to its original position. Label the image of each rigid motion in the provided blanks. a) Rigid Motion 1 Description: Reflection through line (hint sketch in BD above) Label the following square if the following rigid motion was performed on the square shown above it. b) Rigid Motion 2 Description: 90 clockwise rotation around the center of the square. Label the following square if the following rigid motion was performed on the square you got from part a). c) This square shows a rigid motion that took place from your answer in part b. Describe the rigid motion that took place Rigid Motion3 Description:

9 Today s Goal: How can we use rigid motions to map triangles onto other triangles? Mapping Figures 6-2 Notes The idea of pairing each vertex of the to its corresponding in the. How we ll use it: Important Notation! A. (P) B) (P) ( ) Only of degrees are appropriate for question! Fact Check Shortcut methods (SSS, SAS, HL, ASA, AAS) are used to show triangles are. Let's Try It! a) What shortcut method is marked in the triangles to the right? b) Describe a sequence of rigid motion(s) you would use to map ΔABC onto ΔA'B'C' to verify that shortcut method. c) In transformation notation?

10 Mapping with MORE THAN ONE rigid motion: Given DABC and DDEFbelow, DE, DF, ÐD. Describe a sequence of rigid transformations that shows DDEF. Remember, the congruent angles and sides must match! *Plan: a) What shortcut method is marked in the triangles shown right? b) Describe what type of rigid motion(s) you would use to map ΔABC onto ΔA B C to verify that shortcut method. HELPFUL HINTS! 1. Try to get one pair of corresponding points to map. (Make them touch somewhere- typically a translation). 2. Every move should help you map a pair of points! 1 st - 2 nd - 3 rd - 1 st : A of DABC along vector that maps onto C will be called. *note, as long as letters are in right location, you will get credit. 2 nd : a of D A B C about point such that maps onto point C will be called. 3 rd : a of D A B C over (or C ''' B'' ' ) Conclusion statement: yes, you NEED THIS! (Are the two triangles congruent? Explain your reasoning.)

11 Try this one with a partner! Given A D, AC DF, and F, using rigid motions, prove that ABC DEF. Write your thinking in the for each step that would need to occur. Conclusion statement: (Are the two triangles congruent? Explain your reasoning.) Practice Time! 1. Given: a) What shortcut method would prove? b) Describe the rigid motion(s) that would map one triangle onto the other. c) What happens to point A and C in this rigid motion? Why? 2. The following diagram shown pre-image being mapped on to image. The triangles are numbered to indicate which rigid motion they represent in the sequence. A) Rigid Motion (s) used are: 1 2 B) List the corresponding vertices (pre image image ) 3

12 3. Use the diagram shown right to answer a-c: a) Identify each rigid motion that took place (Use details) b) Are the triangles congruent? Justify your answer. 4. Describe a sequence of rigid motions that will map triangle ABC on to triangle A B C

13 5. a) Sketch the segment that represents the line of reflection for quadrilateral and its image. b) What is true about each point on and its corresponding point on? Explain how you know! Remember! What do we know about the line of reflection with respect to the segment formed by connecting the pre-image 7. The figure on the right shows a) Find a sequence of rigid motions that maps one figure onto the other. Be specific! b) Write your answer from part (a) in composition notation.

14 6-2 HOMEWORK Complete each of the following problems and check the key online! Given: The triangles marked below: a) What shortcut method would prove these triangles congruent? b) What is the sequence (just list them!) of rigid motions that prove the two triangles are congruent? ( LOOK AT MARKINGS! 1. Consider the two figures below. Describe a sequence of rigid motion(s) that can be used to prove that. Be specific! ( MARK UP THE CONGRUENT ANGLES AND SIDES FIRST!) 2. Draw in the vector that defines each translation below.(dashed is the pre-image)

15 4. Which transformation is not related to congruence? (1) dilation (2) translation (3) rotation (4) reflection Explain why your choice! 2. Consider the following diagram to answer parts a-c. Given: DE DG and GF State what point is the result of a) r DF D is *This means reflect point D over DF b) r DF E is r c) The rigid motion DF maps F onto 7. Fill in the blank spaces in the appropriate terminology. Indicate each part in your diagram. Let s examine a figure being mapped onto another through a composition of rigid motions. To map we first around. Then across.finally, the second image along the to obtain Since each transformation is a,. We can use notion to describe the composition described as =

16 6-3 Notes Today s Goal: How can we use rigid motions to prove that two figures are congruent in a two column format? Warm-Up 1. Rotate about point c. 2. Translate so that A maps to B. 3. Reflect across segment CD. Think Pair Share! With a partner, read through the following problem and answer the questions that follow Given: C is the image of A after a reflection over BD and ABD CBD ABD and are drawn. Prove using rigid motions: 1. What information do we get from the given? Follow up: This means that BD is also the of AC! 2. Do we see any proof tools in the given? 3. What are we asked to do? Is this similar to anything we ve done before?

17 Let s try it together! 1. Given: C is the image of A after a reflection over BD and ABD and are drawn. Prove using rigid motions: ABD CBD,, Statements 1. C is the image of A after a reflection over and ABD and CBD are drawn. 1. Given Reasons 2. maps onto 2. B is on the line of reflection BD 3. D maps onto 3. D is of reflection BD 4. ABD CBD 4.

18 1. Given: In the diagram below is the perpendicular bisector of and we are given a pair of congruent triangles. Prove: using Rigid Motions.,, Statements 1. CH is the perpendicular bisector of 1. BCE and ABC and DEC are drawn Reasons ABC DEC 7. All corresponding vertices in can be mapped onto a vertex in through NOTE! If you are reflecting over a line, you must be TOLD it is a line of reflection OR State that it is based on the givens.

19 Practice 1. Given with vertices ( ), ( ), ( ) and with vertices ( ), ( ), ( ) as graphed on the set of axes below DEF is the image of ABC after a sequence of transformations. Is Use properties of rigid motions to explain your answer.

20 2... Prove using rigid motions: JYM AYM Think! What type of segment is YM based on the givens? Statements. Reasons Given Definition of a perpendicular Bisector

21 6-3 HOMEWORK 1. Which transformation is not related to congruence? Explain. (1) dilation (2) translation (3) rotation (4) reflection 2. Consider the two figures below.. Describe a sequence of rigid motion(s) that can be used to prove that. Be specific! 3. Assume that the following figures are drawn to scale. Use your understanding of congruence to explain why square ABCD and rhombus GHIJ are not congruent.

22 . Statements Reasons 4. The diagram below shows a pair of congruent triangles, with ADB CDB and ABD CBD. C is the image of A after a reflection over BD. Prove using rigid motions: ABD CBD 1. ADB CDB and ABD CBD. C is the image of A after a reflection over BD Given Given Quadrilateral ABCD is a parallelogram with diagonals AC and BD intersect at E,.

23 6-4 Notes Today s Goal: What is special about angle bisectors with rigid motion proof? Fact Check: In an isosceles triangle, if a perpendicular bisector is drawn in, it is also the of the vertex angle. If we are given : 1. BD is Angle bisector in an isosceles triangle Then we know 2. Which means we also have a 3. Let s try one! How do we know this is a twocolumn rigid motion proof? Given: Isosceles DABC, BC and BD the angle bisector of ÐB Complete the proof using rigid motions: DCBD Statements 1. Isosceles DABC, BC and BD, the angle bisector of ÐB 1. Given Reasons In isosceles triangles perpendicular bisectors and angle bisectors coincide Perpendicular bisectors are lines of reflection 4. AD DC AD DC and All pairs of corresponding vertices map onto each other through rigid motions

24 Non-Negotiable 1 Mapping Name: 6-4 Geometry 2. Describe a sequence of rigid motions that will map. [Be very descriptive in each step!!] Sketch phases if it helps! 1 st - 2 nd - 3 rd Approved by teacher:

25 Non Negotiable 2) Rigid Motion Proof Name: 6-4 Geometry Given: G is the image of E after a reflection over DF and are drawn Prove DEF DGF using rigid motions Statements Reasons Given Approved by teacher:

26 6-4 REVIEW COMPLETE AND CHECK! Bring in booklet to be checked tomorrow! 1. a) List a sequence of rigid motions that will map. (You don t have to describe here.) b) Use the properties of rigid motions to explain why 2. In the diagram of DLAC and DDNC below, DN, CN, and DAC ^ LCN.. Describe a rigid motion(s) that will map DLAConto DDNC.

27 3) Use the diagram shown where LAC DNC for parts A and B. a)fill in: Mapping: The rigid motion R C D maps onto The rigid motion R C N maps onto The rigid motion R C C maps onto 4) In the diagram below, and points A, C, D, and F are collinear on line. Let (not drawn) be the image of after a translation along, such that point D is mapped onto point A. What will the next rigid motion be in this sequence to map D E F onto ABC? Are these triangles congruent? Why or why not?

28 5. Given: Isosceles DABC, BCand BD, the angle bisector of ÐB Prove using rigid motions: DCBD Statements Reasons

29 6. Go to your looking forward and answer your goal problems 2 points! 7. Given: C is the image of A after a reflection over BD are drawn. Prove using rigid motions: ABD CBD Complete the proof: ABD CBD Statements 1. : C is the image of A after a reflection over BD and drawn. ABD and CBD are Given Reasons ABD CBD 4. Hi Geometry Friend! You worked hard! Check your work to see if you did it right!