Proving Figures Are Congruent Using Rigid Motions Using Congruent Are Figures Proving Lesson Lesson 2

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© Houghton Mifflin Harcourt Publishing Company A thatconfirm landscape the elements are congruent. A landscape architect to landscape designthe agrid uses around Use amall. tracing paper to sequence of motions rigid (that is, by asequence of reflections, translations, and/or rotations). Two plane figures are congruent ifand only ifone obtained can be from other the by a Essential Question: 3.2 Name Module 3 Module 1. C B Reflect

the other. The transformation would needto includeadilation, whichisnotarigidmotion. thefiguresIf are notthesamesize, there isnorigidmotionthat maponeofthemonto can that mapsoneto theother. The poolsare congruent there because isarigidtransformation You over NPQR withareflection mapJKLMto can thefold line. transformation that mapsoneto theother. down 4units. The planters are congruent there because isarigid You mapABCD can to to There isnosequence ofrigidtransformations that maps△DEF

How sizes do the of pairs the of figures help are determine ifthey congruent? Explore confirm aboutconfirm lawns? the about confirm this does planters? the tracing that paper so planter ABCD Trace planter atransformation. Describe ABCD you can to use move the transformation that maps △LMN lawns the Determine whether are congruent. Is there arigid aboutconfirm pools? the mapped onto transformation. the NPQR pool . Describe What this does Trace JKLM pools △LMN. The lawns are notcongruent. Using Motions Rigid Proving Figures are Congruent How canyou determine whethertwo figures are congruent? Confirming Congruence and and to to NPQR. Fold that paper the JKLM so pool EFGH withatranslation right 4unitsand

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5/14/14 5:10PM 128 Each rigid Each rigid Yes, if ABCD is Yes, yes, because they have the same because they have yes, How can a rigid motion be used a rigid determine can to motion How congruent? are figures if two Do the figures appear to be congruent? Why Why be to congruent? Do appear the figures not? why or be to the be figure considered Can either not? why or Why preimage? EXPLAIN 1 1 EXPLAIN EXPLORE EXPLORE CONNECT VOCABULARY VOCABULARY CONNECT INTEGRATE TECHNOLOGY INTEGRATE QUESTIONING STRATEGIES QUESTIONING QUESTIONING STRATEGIES QUESTIONING Determining if Figures are Congruent are if Figures Determining Ask students to give examples of of examples give to Define congruence. students Ask might Students in the classroom. figures congruent or shape, floor size and tiles the same with mention size and the same of rectangles are that desktops is shape” size and same “the that students Tell shape. may figures whether two deciding of way informal an congruence of definition a formal but be congruent, basedis rigid motions. on Confirming Congruence Confirming congruence confirming of the option have Students in the book either activity the rigid motion using online. or motion preserves size and shape, so if a sequence of so if a sequence and shape, motion preserves size the to map one figure to motions can be found rigid are and image figure then the preimage other, congruent. size and shape size , then the reverse is true. , then the reverse EFGH to congruent DO NOT EDIT--Changes must be made through "File info" "File through made be must EDIT--Changes NOT DO CorrectionKey=NL-A;CA-A 3/20/14 5:38 PM

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x, Y ( x, ( Z y 0 8 X -8 J -4 CDEF reflectingCDEF JKLM by L -8 You canJKL to map △ △XYZ with a You a by followed reflection the y-axis, across are so the figures translation, horizontal for notation coordinate The congruent. the reflection is the translation is the translation K onto 3. 128 . ) a counter-clockwise rotation of 90° around the origin the of 90° around rotation a counter-clockwise . -y figures are/are not congruent. not are/are figures to JKLM, so CDEF the maps that reflection a rigid is This motion congruent. are figures two rigid transformation that will map one to the other. to one will map that rigid transformation x, ( →

the reflection is for notation coordinate The △ABC to △XYZ, so the two maps that a rigid motion not is/is This . is the rotation for notation coordinate The be to the same/different. appear figures two The △ABC to △XYZ map can You by a so look for shape, be and size to the same appear figures two The CDEF map can You x x ) B A K x y 8 8 J x, ( A W 4 M 4 C X Z Determining if Figures are Congruent are if Figures Determining D Z L y y y 0 0 8 E 0 4 -4 -8 -4 -8 -4 F congruent. answer. Explain your Y B -4 -4 X decide congruence to of the definition whether figures are the two Use -4 Learning Progressions Learning C Y C D -8 -8 -8 You can to map ABCD WXYZ with a You so the figures reflection the x-axis, across notation coordinate The congruent. are the reflection is for

PROFESSIONAL DEVELOPMENT PROFESSIONAL In this lesson students learn that two figures are congruent if and only if there is a is if there only if and congruent are figures two learn that students this lesson In if they means That the other. to figure one maps that rigid motions of sequence then they the other, to figure one maps that rigid find motions can the of sequence if also means, It congruent. are figures image and the preimage confirm that can rigid motions of a sequence is there that be to congruent, known are the figures will use this students lessons, upcoming In the other. to figure one maps that triangles. for criteria congruence develop to definition transformations-based Example 1

2. Module Module 3 Turn Your decide congruence to of the definition congruent. whether figures are the two Use answer. Explain your   1 Explain

GE_MNLESE385795_U1M03L2.indd 128

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CorrectionKey=NL-C;CA-C DO NOT EDIT--Changes must be made through "File info" "File through be made must EDIT--Changes DO NOT CorrectionKey=NL-C;CA-C DO NOT EDIT--Changes mustbemadethrough "File info" maps oneto theother. congruent, there isasequence ofrigid motionsthat toknown becongruent—by thedefinition of that work. Encourage students to for look alternate sequences that maps one figure to another may not unique. be image. Point out that sequence the of motions rigid comparing sizeand the shape of preimage the and esn 3 .2 Lesson 129 plane and on acoordinate plane. congruent figure. Point out thatina both is this true sequence of motions rigid must produce also a motion can produce acongruent figure. Therefore, a motionsto rigid by pointing out how asingle rigid Help students understand how congruence may this when true. be congruent figure. Ask students toexamples find of a sequence of motions rigid to map afigure to a map one figure to other. the Explain that it may take congruent, there then is one motion rigid that can Some students may that think iftwo figures are MP.1 Focus onMath Connections Finding Motions aSequence ofRigid across thex-axisandraises thefigure by 2units. transformation eachx-coordinate reflects figure to other the must exist? CONNECT VOCABULARY AVOID COMMON ERRORS QUESTIONING STRATEGIES PRACTICES INTEGRATE MATHEMATICAL EXPLAIN 2 Relate congruence motion to rigid by ( What would notation the a sequence of motions rigid that maps one For each pair of figures, how do you know that x, y ) → ( -x, y+2 )

mean? If thefiguresIf are The is related CorrectionKey=NL-A;CA-A DO NOT EDIT--Changes mustbemadethrough "File info" GE_MNLESE385795_U1M03L2.indd 129

© Houghton Mifflin Harcourt Publishing Company their owntheir sequences. software motions geometry to use them rigid do the and check results the against sequence of motions rigid to that confirm given the figures are congruent. Have figure to other. the Instruct to them switch papers and another use student’s plane. Have each student asequence of motions rigid describe that map will one Give students coordinates the of apair of congruent figures coordinate inthe Small Group Activity COLLABORATIVE LEARNING Translation: must some sequence be of motions rigid that maps one to other. the The definition of congruence tells you that two when figures are known congruent, to be there Explain2 oue 3 Module 5. tofigure other. the Give coordinate notation for transformations the you use. motions of are figuresshown rigid The Find that congruent. asequence maps one Your Turn 4. Reflect Rotation: Map  Example 2

or rotation willpreserve theoriginalorientation. thetransformationsIf thentheorientation includeareflection, willchange. Atranslation JKLM How is orientation the of figure by the affected asequence of transformations? -8 horizontal translation. of 180°around origin, followed the by a ABC △ABC -8 A Z -4 ≅ WXYZ to △PQRwith arotation△ABC to -4 transformations you use. you transformations that maps one to figure other. the Give coordinate notation for the motions of are figuresshown rigid The Find congruent. asequence ≅ M -4 B ( △PQR -8 -4 x, 8 C 0 ( 4 8 0 y x, ) y W

→ y y Finding Motions aSequence ofRigid ) Y R

4 → ( X Q 4 -x, J ( L x 8 -y + K 8 x x 1, ) P

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→ → ) . 129 followed by a. Map

6.  ABCD ABCD ABCDE -8 : , : -8 reflection acrossreflection they-axis Translation Reflection M to JKLMwith a to ABCD -4 E R ≅ PQRST ≅ Q A -4 JKLM -8 L S J 0 4 8 0 K translation y y D B B P A ( ( 4 4 x, x, C T C y y ) )

→ → 8 8 D x x

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→ → 3/20/14 5:38PM 130 Yes. Yes. Two figures are congruent congruent are figures Two Yes. The angle measures measures angle The Yes. Since rigid motions preserve angle motions rigid Since Can you say two angles are congruent if they congruent are angles two say Can you that the segments but measure the same have if congruent are segments two say Can you Explain. different? is their orientation How does the congruence of angles and and angles of does the congruence How two of the congruence to relate segments How are congruent figures related to to related figures congruent are How transformations? ELABORATE ELABORATE EXPLAIN 3 3 EXPLAIN QUESTIONING STRATEGIES QUESTIONING SUMMARIZE THE THE SUMMARIZE LESSON QUESTIONING STRATEGIES QUESTIONING AVOID COMMON ERRORS COMMON AVOID The orientation of the segments does not affect of the segments orientation The does not affect their and therefore their lengths, congruence. identify the rays that form the angle are different different are the angle form that identify the rays Explain. lengths? figures? Why? Why? figures? determine if the two angles are congruent, not the congruent, angles are if the two determine up their sides. make that or parts of the rays rays Investigating Congruent Segments Segments Congruent Investigating and Angles be cannot angles two believe that may Students have the angles forming if the rays congruent continue rays that students Remind lengths. different a so representing the direction, length in one forever congruent two Draw arbitrary. is in a diagram ray the angles Discuss why rays. longer with one angles, be to larger. appears one though even congruent are if one can be mapped to the other by a rigid a rigid the other by if one can be mapped to reflection, or translation) (rotation, transformation transformations. of rigid a sequence or by measure and distance, verifying that corresponding corresponding that verifying and distance, measure the same have segments angles and corresponding figures two whether determines measure congruent. are DO NOT EDIT--Changes must be made through "File info" "File through made be must EDIT--Changes NOT DO CorrectionKey=NL-A;CA-A 3/20/14 5:38 PM

Lesson Lesson 2 ― EF

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and and two that prove to use transformations Can you ― EF C Investigating Congruent Segments and Angles Congruent Investigating are congruent. The transformation transformation The congruent. are figures? be congruent not but angles congruent have Can figures B Determine which angles or segments are congruent. which Describe segmentsDetermine are angles or can be that used verify congruence. to transformations to either of the other angles. the other of either ∠B to A is a translation. There is no transformation that that transformation no is There a translation. is maps ∠A and ∠C are congruent. congruent. ∠B and ∠C are congruent. In both cases, a sequence a sequence both cases, In congruent. is a reflection and a of transformations translation. Determine which segments and which angles are are angles which and segments which Determine be can that Describe transformations congruent. used the to show congruence. Yes, the definition of congruence for angles requires only that the angle between the rays the rays the angle between only that angles requires for the definition of congruence Yes, does not matter. lengths of the segments The be the same. Essential QuestionEssential Check-In Discussion Can you say two angles are congruent if they have the same measure but the but measure the same if they have congruent are angles two say Can you lengths? different are the angle form that identify the rays that segments figures are not congruent? not are figures 1 maps one figure onto the other, then the figures then the figures the other, with scale If factor onto a dilation ≠ 1 maps one figure Maybe. cannot be congruent. so they cannot be mapped using only rigid motions, not Yes, two figures can have congruent angles but not be congruent figures. They could could They figures. angles but not be congruent congruent can have figures two Yes, of the same figure. versions sized be different appear to

DIFFERENTIATE INSTRUCTION DIFFERENTIATE Draw two congruent triangles and label the vertices. Highlight one side of one one of side one label the vertices. Highlight triangles and congruent two Draw triangle and the other of side the corresponding name students Have triangle blue. corresponding of pairs two the other with Repeat well. as blue side that highlight students Have red. the angles of one shade Then purple. and green using sides well. as red that shade triangle and the other of angle the corresponding name orange. and yellow using angles corresponding of pairs two the other with Repeat Visual Cues Visual Example 3

7.  Module Module 3 9. Elaborate Elaborate 8. Turn Your congruent 3 Explain are angles Two figures. whole as well as figures parts of to refer Congruence can of a sequence by is, (that rigid motions by the other from be can obtained if one only if and segments two for required are conditions same The rotations.) and/or translations, reflections, other. each to beto congruent 10.

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CorrectionKey=NL-C;CA-C DO NOT EDIT--Changes must be made through "File info" "File through be made must EDIT--Changes DO NOT CorrectionKey=NL-C;CA-C DO NOT EDIT--Changes mustbemadethrough "File info" esn 3 .2 Lesson 131 and segment lengths are preserved. measuringthe features to show that angle measures sequence of motions. rigid Remind students to use congruent by software using geometry to do a Focus on TechnologyFocus MP.5 PRACTICES INTEGRATE MATHEMATICAL ASSIGNMENT GUIDE EVALUATE and Angles Investigating Congruent Segments Example 3 Finding Motions aSequence ofRigid Example 2 Congruent Determining ifFigures are Example 1 Confirming Congruence Explore Concepts andSkills Students that can verify two figures are Practice 23, 25 Exercises 10–13, 18–21, 24,26–31 Exercises 6–9,16, 14–15, 17 Exercises 1–5, CorrectionKey=NL-B;CA-B DO NOT EDIT--Changes mustbemadethrough "File info" GE_MNLESE385795_U1M03L2 131

Exercise © Houghton Mifflin Harcourt Publishing Company 24–29 10–23 1–5 6–9 5. 3. 1. yourExplain answer. Give coordinate notation for transformations the you use. Use two the figuresare whether congruent. definition the ofto congruence decide oue 3 Module are congruent. reflection: horizontal translation. So, thetwo figures △CDE over thex-axis, followed by a You △JKLby map△CDEto can reflecting translation: Q translation: translation. So, thefigures are congruent. You PQRST witha mapABCDEto can Evaluate: Homework andPractice -8 -8 P Q E S -4 -4 2 1 1 1 Depth ofKnowledge (D.O.K.) B R P Skills/Concepts ofInformation Recall of Information Recall of Information Recall R C C D A -8 -4 -8 -4 -8 -4 4 8 4 8 4 0 0 0 ( ( x, D y x, y y L y T y E ) ) 4 4

E K → S J → B D ( ( 8 8 8 x x C x x x + 8,y - 2,y ( x, are notcongruent. that willmaponefigure onto theother, sothey There isnosequence ofrigidtransformations y ) - 7 . )

→ ) . ( x, -y ) ; 131 MP.5 MP.5 MP.4 MP.5 COMMON 4. 2. CORE are congruent. reflection: horizontal translation. So, thetwo figures acrossreflection thex-axis, followed by a You mapWXYZ can translation: UsingTools UsingTools Modeling UsingTools congruent. figure onto theother, sothey are not transformations that willmapone There isnosequence ofrigid -8 -8 Z X W Mathematical Practices -4 -4 Z Y -8 -4 -8 Y 8 4 8 ( 0 0 x, X M y y E L y ) D

→ 4 4 to to ( G DEFG x 8 8 + 10,y F N x x ( x, with a y • Extra Practice Extra • Help and Hints • Homework Online • ) )

→ . (

x, esn 2 Lesson -y ) ; 5/14/14 5:15PM 132 Suggest that students use tracing paper to to paper use tracing students that Suggest INTEGRATE MATHEMATICAL MATHEMATICAL INTEGRATE PRACTICES Focus on Modeling Focus MP.4 investigate which corresponding segments for two two for segments corresponding which investigate of pairs which and congruent are figures them fold Have congruent. are angles corresponding coincident see to are paper if the figures the tracing rigid of a sequence then write if they are, and, Have the other. to figure one map can that motions the figure them draw also to use paper the tracing then and paper graph on image congruent its and the to figure one map rules that the algebraic give other. DO NOT EDIT--Changes must be made through "File info" info" "File through made be must EDIT--Changes NOT DO CorrectionKey=NL-A;CA-A 3/20/14 5:38 PM

) 6 -x, -x, ( (

+ ) y y → →

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4 ) ) E y y y Mathematical Practices Mathematical y x, x, ( (

0 0 B D 8 4 Reasoning Reasoning X Modeling -4 -8 -4 -8 △WXY △DEF F CORE ≅ ≅

-4 -4 COMMON with a rotation of △ABC toDEF with a rotation △ △CDE to △WXY of 180° with a rotation MP.2 MP.2 MP.4 are all congruent. all congruent. ∠B and ∠C are W D Y B -8 -8 ∠A, is a transformations of sequence The reflection and a translation. △CDE △ABC around the origin, followed by a by the origin, followed 180° around rotation: translation. Map translation: translation: translation: translation: Map a horizontal by the origin, followed around rotation: translation. 9. 7. 11. 132 ; ; ) ) y y

) -x, -x, 4 ( (

- ) → →

y 8 ) ) y y 1, - y x, x, x + X Y x ( ( D H x, x 8 8 ( (

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4 ) ) Z B y y Strategic ThinkingStrategic W Thinking Strategic Thinking Strategic y y Q S x, x, Depth of Knowledge (D.O.K.) Depth of Knowledge 3 3 3 ( (

0 E 0 G 8 4 C 8 4 PQRST -4 -8 R U WXYZ ≅ R ≅ -4 -4 to PQRST with a reflectionDEFGH RSTU to WXYZ with a reflection 36 S T 34–35 30–33 T P A -8 DEFGH Map a vertical by followed the y-axis, across reflection: translation.

RSTU -axis, followed by a by followed the y-axis, across reflection: translation. Map translation: translation: translation: translation: None of the angles are congruent. There There congruent. None of the angles are maps one of that is no transformation another. the angles to Exercise

8. 10. Determine which of the angles are congruent. Which transformations can be used congruent. transformations Which which the angles are Determine of verify theto congruence? 6. The figures shown are congruent. Find a sequence of rigid motions that maps one one maps a sequence congruent. that Find The rigid shown figures are of motions use. you the transformations for notation coordinate Give the other. figure to Module 3

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CorrectionKey=NL-C;CA-C DO NOT EDIT--Changes must be made through "File info" "File through be made must EDIT--Changes DO NOT CorrectionKey=NL-C;CA-C DO NOT EDIT--Changes mustbemadethrough "File info" esn 3 .2 Lesson 133 color, using adifferent color for each pair. highlight pairs of corresponding sides same inthe each pair of corresponding Students vertices. can also same inthe vertices color, using adifferent color for its corresponding vertex by circling corresponding paper, students how can see each vertex is mapped to Focus onMath Connections MP.1 PRACTICES INTEGRATE MATHEMATICAL When examining congruent figures on graph CorrectionKey=NL-C;CA-C DO NOT EDIT--Changes mustbemadethrough "File info" GE_MNLESE385795_U1M03L2.indd 133

© Houghton Mifflin Harcourt Publishing Company 16. 14. yourExplain answer. Give coordinate notation for transformations the you use. Use two the figuresare whether congruent. definition the ofto congruence decide 12. tocongruence? the verify used of areDetermine segments Which the transformations which congruent. be can Module 3 Module translation. rotation: around theorigin,followed by avertical counter-clockwise rotation of90° Yes. MapEFGH RSTUwitha to translation: AB by atranslation. rotation: rotation of90°around theorigin,followed Yes. translation: maps translation. There isnotransformation that ― -8 G X F B and

W F withaclockwise △WXY Map△JKLto -4 -4 EF

― CD ― A to eitheroftheothersegments.

C J -4 E H 4 8 are congruent; then reflection,

4 8 y Y 0 0 (

(

E x, x, R S y D y y ) ) 4

L K → →

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( ( 8 8 x, x x, T U x x + y y + ) ( 1,

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133 17. 15. 13. . △WXY △KLM to There are notransformations to map No, thefigures are notcongruent. maps oneofthemto another. There isnorigidtransformation that None ofthesegments are congruent. translation: reflection: reflection: by ahorizontal translation. acrossreflection thex-axis, followed Yes. -8 -8 A D K K L to JKLMNwitha MapBCDEFto -4 L C D -8 -8 M C 4 8 4 8 X

( 0 0 x,

( M x, B y y y ) y E E

) → N 4

F W → J

( x,

( 8 8 F x B -y Y x x - ) 4,

y ) . Lesson 2 Lesson 6/9/15 12:00AM 134 AVOID COMMON ERRORS COMMON AVOID Students may make an error when using when using error an make may Students is figure if a transformed determine to computations a that Emphasize congruent. not or congruent is other each crossing the sides with figure resulting a necessarily not error, an of indication an figure. noncongruent DO NOT EDIT--Changes must be made through "File info" info" "File through made be must EDIT--Changes NOT DO CorrectionKey=NL-C;CA-C 6/9/15 12:00 AM

D Proving Figures are Congruent Using Rigid Motions Using Congruent are Figures Proving ; ; A ) ) Lesson 2 y -y -x, (

-x, (

. . ) ) →

6 y ) →

y ) 4, + y B x, y - ( x x x, K x x, ( 8 (

(

J C with a reflection L F → M →

4 ) ) D y y JKLM y y JKLM. x, Y x, (

(

to 0 0 L to 88 8 4 to △KLM. -8 -4 Z WXYZ with a rotation of 180° △DEF to △KLM with a rotation △DEF WXYZ WXYZ W K X Not congruent Not congruent Not congruent Not congruent Not congruent E M -8 -8

Map Map Map translation: translation: translation: Map a horizontal by about the origin, followed Rotation: translation. -axis, followed by a vertical by followed the y-axis, across reflection: translation. 21. 19. 134 . ) . ) Congruent Congruent Congruent Congruent Congruent

-y y + 10 -x, (

→ x + 6, x

x E D (

E ) 8 with a rotation of 180° with a rotation 8 y H → F x,

( ) AA 4 y with a combined ABCDEF with a combined D S T . ABCDEF x, DEFGH DEFGH C B ( F . DEFGH y to y G to to to 0 0 8 8 4 S R U P -8 P R Q PQRSTU PQRST PQRSTU PQRSTU PQRST PQRST ∠A and ∠B ∠A and ∠C ∠B and ∠C ∠B and ∠D ∠C and ∠D T

Q -8 a. b. c. d. e. -8 Determine whether each pair of angles is congruent or not congruent. congruent. not or congruent is angles of whetherDetermine pair each part. lettered each for Select answer the correct Map Map Map Map translation. The coordinate notation for the for notation coordinate The translation. is translation around the origin. The coordinate notation notation coordinate The the origin. around is the rotation for

22. 20. Module 3 18. The figures shown are congruent. Find a sequence of transformations for the for of transformations a sequence Find congruent. are shown figures The use. you the transformations for notation coordinate Give mapping. indicated

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CorrectionKey=NL-C;CA-C DO NOT EDIT--Changes must be made through "File info" "File through be made must EDIT--Changes DO NOT CorrectionKey=NL-C;CA-C DO NOT EDIT--Changes mustbemadethrough "File info" esn 3 .2 Lesson 135 its corresponding motions. rigid pile. Students to match should try each drawing with papers that so each student receives one from each and one pile of descriptions. Randomly pass out the sheets, half making the one pileCollect of drawings language or symbols, cut then paper the inhalf. sequence of motions rigid using precise mathematical only once. bottom On the half, have write them the draw may A'B'C'.They each transformation use perform asequence of two or motions three rigid to half, have students draw. Then ask ABC to them Give each student asheet of graph paper. top On the COLLABORATIVE LEARNING CorrectionKey=NL-B;CA-B DO NOT EDIT--Changes mustbemadethrough "File info" GE_MNLESE385795_U1M03L2 135

© Houghton Mifflin Harcourt Publishing Company 23. Module 3 Module 28. 26. 24. sequence oftransformations mustincludeareflection. False. The figures donothave thesameorientation, sothe B. C. If If A. Which sequence of transformations not does map afigure onto acongruent figure? Explain. for transformations the you use. CDEFG QRSTU to . Give coordinate notation sequence of transformations that map will The figures shown are congruent. Find a congruent? Explain. Which segments are congruent? Which are not or Then false. explain your reasoning. using arotation and atranslation. statement the Determine whether is true D. -8 C F

WXYZ and ABCD are mapped congruent, can be ABCD then toWXYZ

Reflection across y-axis, combined the Reflection translation Rotation of 180°about Rotation of 180° translation ( y-axis, combined translation Counterclockwise rotation of 90°about x, -4 A y ) Q U

-8 → 4 8 0 (

2x, y y T (

x, D E C D G y 4 ) B )

about →

F E S R G A dilation isnotarigidtransformation. (

8 x H x the origin, reflection across origin, reflection the y-axis, the dilation the origin, reflection across origin, reflection the x-axis, the horizontal + 4, y Then translate: across thex-axis: Then reflect about theorigin: 90° clockwise Rotate ( ( ( onto another. one segment motions map No rigid congruent. None are x, x, x, y ( - 9 ) x,

y y y ) ) ) y )

) → → →

. → CDEFG

( ( ( x x, y, (

the origin, reflection across origin, reflection the the x - 4, -x -y -11, ) ) 135 . .

y 27. 25.

- 12 ( x, the transformationsthe you use. △XYZ △LMN to sequence of transformations that map will The figures shown are congruent. Find a congruent? Explain. Which angles are congruent? Which are not -8 Z Y y )

)

→ -4 (

x D -4 -5, X 4 8 0 B A y y C + . Give coordinate notation for

4 2 N L )

D B 8 A M x X y (x, Translate: (x, y-axis: across the Reflect (x, about theorigin: counter Rotate onto another. map oneangle rigid motions congruent. No None are - 13). C y) y) y) Y W → (x → (-x, → (-y, △LMN 90° clockwise Lesson 2 Lesson - 3, y). x). Z 5/14/14 5:41PM ” ) 136 x, −y ( → ) x, y ( Be sure that students’ answers are as detailed as are answers students’ Bethat sure INTEGRATE MATHEMATICAL MATHEMATICAL INTEGRATE PRACTICES MP.3 Focus on Communication Focus and precise as possible. When stating that two figures figures two that stating When possible. as precise and students congruent, are plane in the coordinate describe map should the specific that rigid motion(s) reflection “a instance, For the other. to figure one reflection “the or the x-axis” across are detailed descriptions of a rigid motion in the a rigid motion of detailed descriptions are not. is while “reflection” plane, coordinate DO NOT EDIT--Changes must be made through "File info" "File through made be must EDIT--Changes NOT DO CorrectionKey=NL-A;CA-A 5/14/14 5:41 PM

© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©ImageClub/Getty Images; ©Atlaspix/Shutterstock x D x 8 Proving Figures are Congruent Using Rigid Motions Using Congruent are Figures Proving 8 Lesson Lesson 2 G 4 C E F y y Y 0 8 R S 0 X V 8 4 F -4 -8 P Z W -4 T -4 D E Q -8 H -8 G . ) 9 - y around around 2, ; ) + x ( -x y, The petals can be mapped The a reflection, each other by onto which is a rigid transformation. → ( ) y 136 → x, ) ( y x, ( . Are both students correct, is is correct, both students . Are ) 2 ; translation: ; translation: . The second student believes the correct believes the correct student second . The ) x, y - )

( 6 →

-x, y ) ( Two students are trying to show that the two the two that trying are show to students Two x, y + ( x, y ( → ) →

y ) x, ( x, y ( to VWXYZDEFGH of 90° rotation with a clockwise reflection: to CDEFG map decides to student first The congruent. are figures the by the followed origin, 180° around of a rotation using PQRST translation Draw Conclusions Conclusions Draw The city of St. Louis was settled by the French in the mid 1700s and joined the United the United joined 1700s and in the mid the French settled was Louis by St. of city The flag city The reflects French its Purchase. the Louisiana part in 1803 as of States petals right the left and that prove you can How the fleur-de-lis. featuring history by other? each to congruent are -axis, followed followed the y-axis, a reflection across by the origin, followed rotation: translation. a combined by How can you prove that two arrows in the recycling symbol are congruent to to congruent are symbol in the recycling arrows two that prove you can How other? each Map The figures shown are congruent. Find a sequence of transformations transformations of a sequence Find congruent. are shown figures The the for notation coordinate . Give to VWXYZ DEFGH will map that transformations you use. -axis, followed by the by followed the y-axis, a reflection across are transformations translation vertical Only the first student is correct. The two figures have the same have figures two The is correct. Only the first student including a single of transformations so a sequence orientation, of the result. reflection will change the orientation The arrows can each be mapped to each other by a rotation, a rotation, each other by can each be mapped to arrows The which is a rigid transformation. only one student correct, or is neither student correct? student neither is or correct, student one only

Module Module 3 32. 31. 30. 29.

GE_MNLESE385795_U1M03L2 136

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33. Justify Reasoning Two students are trying to show that the two figures are PEERTOPEER DISCUSSION congruent. The first student decides to map DEFG to RSTU using a rotation of 180° about the origin, followed by the vertical translation ( x, y) → ( x, y + 4) . The second Have students discuss with a partner how to predict student uses a reflection across the x-axis, followed by the vertical translation ( x, y) → ( x, y + 4) , followed by a reflection across the y-axis. Are both students the sequence of rigid motions that may map a figure correct, is only one student correct, or is neither student correct? to a congruent figure. Then have them predict and Both students are correct. Either of the R y test whether reversing the order of the sequence of sequences of transformation will map 8 DEFG to RSTU. Recall that a rotation of S rigid motions will produce the preimage figure. 4 180° around the origin is the same as a T G reflection across both axes. U F x 8 4 0 8 - - E 4 CONNECT VOCABULARY - D Have students relate the word congruent to the terms -8 equal and equivalent. If a figure is congruent to

another it is equal in shape and size. Show students H.O.T. Focus on Higher Order Thinking similar figures that are equal in shape but not size, 34. Look for a Pattern Assume the pattern of congruent squares shown in the figure and discuss why they are not congruent. continues forever. y 4 Write rules for rigid motions that map square 0 0 onto square 1, square 0 onto square 2, and square JOURNAL 0 onto square 3. 1 x, y x 2, y 2 x ( ) → ( + - ) 0 Have students write a journal entry in which -4 -2 2 4 ( x, y) → (x + 4, y - 4) x, y x 6, y 6 -2 they summarize what they know so far about ( ) → ( + - ) 3 congruence. Prompt students to include examples Write a rule for a rigid motion that maps square 0 -4 onto square n. and non-examples of congruent figures and the (x, y) → (x + 2n, y - 2n) methods of obtaining them and determining them. 35. Analyze Relationships Suppose you know that △ABC is congruent to △DEF and that △DEF is congruent to △GHJ. Can you conclude that △ABC is congruent to △GHJ? Explain. Yes; by the definition of congruence, there is a sequence of rigid motions that maps △ABC onto △DEF and another that maps △DEF onto △GHJ. The first sequence followed by the second sequence maps △ABC onto △GHJ, so the triangles are congruent. 36. Communicate Mathematical Ideas Ella plotted the points A(0, 0), B(4, 0), and _ _ C(0, 4). Then she drew AB and AC . Give two different arguments to explain why the segments are congruent. © Houghton Mifflin Houghton © Company Harcourt Publishing Both segments are 4 units long. Because the segments are the same length, they are congruent. A rotation of 90° maps AB¯ onto AC¯ . Because there is a rigid motion that maps one segment onto the other, the segments are congruent.

Module 3 137 Lesson 2

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137 Lesson 3 . 2 138 The The Ask students to show how each of the nine the nine of each how show to students Ask certain object viewed types An through of The letters C, O, D, and E are symmetric with and E are D, C, O, letters The INTEGRATE MATHEMATICAL MATHEMATICAL INTEGRATE PRACTICES MATHEMATICAL INTEGRATE PRACTICES MP.4 MP.3 Focus on Modeling Focus Thinking on Critical Focus pieces in the Lesson Performance Task can be can divided Task in the Performance Lesson pieces shape the entire so that shapes congruent two into shapes. congruent 18 be from can constructed Ask be to flipped upside-down. willlenses appear flipped when are STAR in the word the letters why is CODE the word but a lens such seen through not. shapes are trapezoids that form an L when placed placed an L when form that trapezoids shapes are together. respect to a line drawn horizontally through their through horizontally respect a line drawn to The not. R are A, and T, S, while the letters centers, the lens, is indeed “flipped” by CODE is that result the lens appears exactly and the image through same is not true of the letters The as it did before. of STAR. Scoring Rubric his/her reasoning. and explains the problem solves correctly Student 2 points: but does not fully good understanding of the problem shows Student 1 point: his/her reasoning. or explain solve understanding of the problem. does not demonstrate Student 0 points: 6/9/15 12:00 AM

EXTENSION ACTIVITY EXTENSION The broken lines on the figure show how it can be divided into three congruent congruent three be can into divided it how show the figure on lines broken The can it how determine and the figure copy students isosceles triangles. Have right trapezoids. congruent eight be into divided

Describe how each of Shapes 2–9 can be formed from Shape #1 through a combination of of a combination #1 through Shape from be 2–9 can formed Shapes Describe of each how eight least at using this one, like design a figure Then rotations. and/or reflections, translations, from be them can formed of each describe Then how the shapes. Number shapes. congruent rotations. and/or reflections, translations, of a combination #1 through Shape The illustration shows how nine congruent shapes can be fitted together to form a larger a larger form to together be can fitted shapes congruent nine how shows illustration The translations, of a combination #1 through Shape from be can formed the shapes Each of shape. rotations. and/or reflections, Lesson Lesson Task Performance Module 3 Shape #2: Rotate Shape #1 180°. Shape #2: Rotate Shape #3: ReflectShape #1 vertically. Shape and right. #1 down Shape Translate #4: Shape and right. it down #1 180° and then translate Shape #5: Rotate Shape #1 down. Shape Translate #6: Shape and right. #1 down Shape Translate #7: Shape and then reflect it horizontally. #1 down Shape Translate #8: Shape and right. #1 down Shape Translate #9: Shape answers. Check students' will vary. Answers

GE_MNLESE385795_U1M03L2.indd 138

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