CorrectionKey=NL-C;CA-C DO NOT EDIT--Changes mustbemadethrough “File info” LESSON esn 3 . Lesson 139 or noncongruent. Have students fillinsentence stems to explain why figures are congruent Objective Language Practices Mathematical corresponding pairsofanglesare congruent. two trianglesare congruent ifandonlycorresponding pairsofsidesand Use thedefinitionofcongruence interms ofrigidmotionsto show that The student isexpectedto: Common Core Math Standards Congruent Congruent Figures Are of Parts Corresponding Then preview Lesson the Performance Task. students to identifycongruent shapes design. inthe View online the Engage. photo the Discuss and ask distances between vertices, are equal. relationships withinthe figures, suchasrelative The are corresponding congruent, parts and Essential Question: congruent? conclude abouttwo figures that are PERFORMANCE TASK PREVIEW: LESSON ENGAGE COMMON COMMON CORE CORE MP.2 Reasoning G-CO.B.7 3 . 3 3.

What canyou CorrectionKey=NL-A;CA-A DO NOT EDIT--Changes mustbemadethrough "File info" GE_MNLESE385795_U1M03L3.indd 139

© Houghton Mifflin Harcourt Publishing Company A You investigate will some conclusions you can make you when know that two figures are congruent. Essential Question: 3.3 Name Module 3 Module C B

D

A translation maps eachsideorangleof△ The are corresponding congruent parts there because isasequence ofrigidmotionsthat Explore

and △ DEF and △ ABC them Label triangles. Then cut out triangle, the cutting through layers both of paper to produce two congruent Fold asheet of paper inhalf. Use astraightedge to draw on atriangle folded the sheet. parts ofparts △DEF The same sequence of motions rigid that maps △ DEF to △ ABC motions.rigid must asequence of motions be rigid that maps △ DEF to △ ABC Place next triangles the to each other on adesktop. Since are triangles the congruent, there What Step does Ctell you about corresponding the of parts two the Why? triangles? AB A _

→ → A of Congruent Figures Corresponding Parts

DE ¯ D re Congruent

. Complete following. the ( What canyou conclude abouttwo figures that are congruent? perhaps followed by arotation Exploring Congruence of ofParts Transformed Figures B BC _

→ → EF

¯ E

DO NOT EDIT--Changes must be made CorrectionKey=NL-A;CA-A ABC to thecorresponding sideorangleof△

Name Corresponding 3Parts. 3

Essential Question:

COMMON Explore GE_MNLESE385795_U1M

CORE

You will investigate some conclusions you can mak 

Fold a sheet of paper in hal

G-CO.B.7 Use the definition of congruence in terms of rig congruent if and only if corresponding pairs of congruent.  Then cut out the triangle, cu © Houghton Mifflin Harcourt Publishing Company triangles. Label them △ ABC of Congruent Figures  A Place the triangles next to ea , as shown.

 must be a sequence of rigid mo The same sequence of rigid moti rigid motions. re Congruent A translation

through "File info" parts of DEF 03L3 139 Module 3 △ What does Step C tell you a AB _ A What can you conclude about two figures that are con

The corresponding parts ar C AC maps each side or angle of

_ → Exploring Congruence of Parts of Transformed → Figures Class

DE ¯

D

( . Complete the following. perhaps followed by a ro

f. Use a straightedge to draw a tr

tting through both layers of pape

and △ DEF BC _ B

ch other on a desktop. Since the Class

→ → → →

tions that maps △ ABC to △ DEF

bout the corresponding parts of th EF ¯

ons that maps △ ABC to △ DEF , as shown. E e when you know that two figures are congruent. sides and corresponding pairs of angles are 139 e congruent because there is a 139 △

ABC to the corresponding sid

id motions to show that two triangles are

AC _ C tation

→

→

DF

C ¯ iangle on the folded sheet. Date )

maps ABC to DEF. DF △ △ ¯ A r to produce two congruent )

F 139 F

gruent? triangles are congruent, there

B △DEF. maps△ABCto

F . Describe the sequence of

maps parts of △ ABC to e two triangles? Why? D C

sequence of rigid motions that E

e or angle of △

Resource

Locker A

DEF.

Lesson 3

maps of to parts △ ABC . Describe the sequence the of . Describe HARDCOVER

01/04/14 10:58 PM B F edition. hardcover student find thislessoninthe Turn to thesepagesto edition. hardcover student find thislessoninthe Turn to thesepagesto Date D

E PAGES 123132

DEF. Resource Locker Lesson 3 Lesson 04/04/14 7:00PM

140

6; 3 angles and , an if-and-only-if if-and-only-if , an for general figures and and figures general for is a biconditional is When you are given two congruent congruent two given are you When corresponding of pairs many triangles, how Have a student read the statement read student a Have about EXPLORE EXPLORE 1 EXPLAIN QUESTIONING STRATEGIES QUESTIONING MATHEMATICAL INTEGRATE PRACTICES 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 sides parts—angles and sides—are there? there? sides—are and parts—angles MP.3 Exploring Congruence of Parts of Congruence Exploring Figures Transformed Parts of Congruent Corresponding Congruent Are Figures on Communication Focus Corresponding Parts of Congruent Figures. Discuss Figures. Congruent of Parts Corresponding the statement of the meaning that triangles. Emphasize two of then in terms the statement if-then true is an as when read that statement direction. in either statement 4/5/14 2:04 PM

© Houghton Mifflin Harcourt Publishing Company E R Lesson Lesson 3 65° Q D 73° Congruent are Figures of Congruent Parts Corresponding 42° 3.5 cm P F S C L K °. 3.7 cm A 65 . The rigid motions that map motions that rigid The ∠F. = ≅ B J 2.6 cm ∠C M Corresponding Parts of Congruent Congruent of Parts Corresponding is a biconditional, a statement that is true is that a statement a biconditional, is ∠E, 140 °, so m∠B ≅

. .

65 ∠B

_ E . DE =

_ DE ∠ ∠D, E ≅ ≅

≅ , what six congruence statements about about statements congruence six , what _ AB ∠B , ∠A , = 2.6 cm, △ DEF ¯ DF, . Find the given side length or angle measure. the side given or . Find length , and m∠ , and △DEF △DEF ≅ ≅ E ≅ ≅ DEF AC ¯ △ Are Congruent Are Corresponding Parts of Congruent Figures Figures Partsof Congruent Corresponding ≅ ¯ EF, = 2.6 cm. , and AB AB, and △ABC

△ABC = m∠ ≅ can congruent, JKLM and are PQRS =

DE ABC BC ¯ DE △ m∠B Find the side that corresponds to to corresponds that the side Find length. the unknown Find measure. angle the unknown Find ∠B. to corresponds that the angle Find

¯ DE, Math Background Math ≅ Step 1 Step 2 Step 2 Step 1 DE to the corresponding sides and sides the corresponding △ABC to △ABC to sides and angles of △DEF also map the you make any conclusions about corresponding corresponding about conclusions any make you not? why or parts? Why Do your findings in this Explore apply to figures figures to apply this in Explore findings Do your that know if you instance, triangles? For than other quadrilaterals If you know that △ ABC that know you If Why? write? you can angles and segments , there is a sequence of is a sequence PQRS, there quadrilateral to JKLM is congruent quadrilateral since Yes; of rigid motions maps sides same sequence This maps JKLM to. PQRS rigid motions that PQRS. sides and angles of the corresponding and angles of JKLM to ¯ AB , which establishes congruence. DEF, which establishes angles of △ m∠B Corresponding Parts of Congruent Figures Are Congruent Are Figures Parts of Congruent Corresponding If two figures are congruent, then corresponding sides are congruent and and congruent are sides then corresponding congruent, are figures two If congruent. are angles corresponding

PROFESSIONAL DEVELOPMENT PROFESSIONAL Figures Are Congruent. This statement This Congruent. Are Figures pairs corresponding and sides of pairs if corresponding is, That direction. in either congruent. are then the figures congruent, are figures in two angles of In this lesson, students learn that if two figures (including triangles) are triangles) are (including figures if two learn that students this lesson, In of angles of pairs corresponding and sides of pairs then corresponding congruent, of definition the rigid-motion from readily follows This congruent. are the figures the statement from and congruence that Example 1

so

Since Since Module 3

  The following true statement summarizes what you discovered in the in Explore. discovered you what summarizes true statement following The Explain 1 2. 1. Reflect

GE_MNLESE385795_U1M03L3 140

CorrectionKey=NL-A;CA-A

DO NOT EDIT--Changes must be made through "File info" "File through made be must EDIT--Changes NOT DO

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” the otherletters correspond inthesameorder. letters correspond, thelastletters correspond, and of letters inthecongruence statement. The first congruent. figures are congruent, thenthefigures are ofcongruentapplies becauseifcorresponding parts congruent.corresponding parts Then thestatement could usetransformations to create allpairsof easily answereasily questions. the represented corresponding the can more they parts, figures to show have Once them. they clearly that relate of parts the figures the and mark the esn 3 . Lesson 141 MP.4 Focus onModeling Congruence of Applying theProperties and non-examples of congruent figures poster. inthe areparts congruent. Have show them examples both to labeled showbe which pairs of corresponding concept of congruent figures. The illustrations should Have each student make aposter illustrating the VISUAL CUES QUESTIONING STRATEGIES QUESTIONING STRATEGIES PRACTICES INTEGRATE MATHEMATICAL EXPLAIN 2 Suggest that students congruencies the list all congruent figures correspond? How do you determine which sides of two whether twowhether figures are congruent? How could you transformations use to decide Use theorder You CorrectionKey=NL-B;CA-B DO NOT EDIT--Changes mustbemadethrough "File info" GE_MNLESE385795_U1M03L3 141

© Houghton Mifflin Harcourt Publishing Company parts ofparts figures. the statements that fit pairthe of figures, and list congruentthe pairs of corresponding have switch them papersmore several times within groups, write congruence new switch papers and to write acongruence statement for pair the of figures. Then Have each student draw apair of congruent figures on paper. Instruct to them Small Group Activity COLLABORATIVE LEARNING Since same length, so motionsRigid and length preserve angle measure Explain2 △STU Your Turn 3. Reflect oue 3 Module  So, So, Therefore, angles have same the measure, ∠J so fo ahsd. 8=2x Divide each side by 2. Subtract 3xfrom each side. Write an equation. U Example 2 Symmetric Property ofCongruenceSymmetric Property Reflexive Property ofCongruenceReflexive Property Transitive ofCongruence Property Properties ofCongruenceProperties ht that Discussion

AB J

≅ JK _ AB . Find the given side length△VWX. Find or given the side measure. angle

K

△ABC ≅ = 3x △ABC QR QR UV _ _

AB The shown triangles figure inthe are congruent. Canyou conclude L ? Explain.

≅ 124° 3(4) + 8= ≅ = XY XY _ ≅ △DEF Applying ofCongruence theProperties

DE

Q P implies △DEF T

16 ft . S , AB _ + 8=1220in. V . Find the given side length. Find or given the side measure. angle UV

≅ W 38° ≅ = DE DE _ ∠K implies m∠J XY . 3x

and versa. vice In same the way, congruent 32 ft + 8=5x f If

AB f If _ 4=

AB AB R 43 ft _ _ ≅

. This means that congruent segments have the

≅ ≅ 18° AB _ x CD CD _ _ so you assume cannot correspondence between thetrianglesisnotgiven, No; thesegments appearto becongruent, butthe parts.

= and , then 141

m∠K and versa. vice (3x +8)in. CD _ X CD _ A

(5y +11)° ≅

≅ EF _ SU Since m∠S Since 4. 5. AD _ , then

m∠S SU = .

= VX △STU △STU B AB _ m∠V = 43ft.

≅ ¯ and JK 25 in. EF _ ≅ ≅ = .

△VWX, △VWX, 38°. C ¯ QR are corresponding D

¯ SU (5x) in. ∠S (6y +2)° ≅

≅

¯ VX. ∠V. 83°

E esn 3 Lesson F 5/14/14 6:03PM 142 Corresponding Corresponding Encourage students to use geometry to software students Encourage EXPLAIN 3 3 EXPLAIN CONNECT VOCABULARY VOCABULARY CONNECT AVOID COMMON ERRORS COMMON AVOID MATHEMATICAL INTEGRATE PRACTICES MP.5 learn the Corresponding students this lesson, In Parts . Although Congruent Are Figures Congruent of some to be helpful may CPCTC) as (such acronyms or postulates, statements, to when referring students difficult for more be a bit devices may such theorems, Consider level. the Emerging Learners at English a copy or create students having or a poster making them for their meanings, with along theorems, of list come to want may Students in this module. to refer to as such itself, the CPCTC for a mnemonic with up Carefully. Too Cooks Carrots Pick Students may correctly solve for a variable but then but a variable for solve correctly may Students a side as the variable of the value give incorrectly examine them to Remind measure. angle or length carefully; or the diagram length a side sometimes expression described is an by measure angle alone. the variable by not a variable, containing Congruent Using Parts in a Proof Focus on Technology to reflect the triangle with the given conditions and and reflectto conditions the triangle the given with parts congruent corresponding verifythen to that equal measure. have DO NOT EDIT--Changes must be made through "File info" "File through made be must EDIT--Changes NOT DO CorrectionKey=NL-A;CA-A 04/04/14 7:02 PM

© Houghton Mifflin Harcourt Publishing Company C Lesson Lesson 3 = 98° = m∠M. ° Congruent are Figures of Congruent Parts Corresponding ) y D A quadrilateral ≅ quadrilateral ⋅ 9 + 17 9 ( = ° ) - 1 → 9 = . Therefore, m∠H ∠M. Therefore, L M (6x − 13) cm + 17 ° ≅ ° 9y m∠ D. B ( ∠H (10y) = = Reasons 18 cm + 17 = 11y A (11y - 1) + 2 + 2 P

m∠H LMNP, Since quadrilateral GHJK quadrilateral Since 9y m∠H y y 6y N ° . 7. 142 Given Corresponding parts Corresponding of congruent congruent. are figures Definition of midpoint. = = =

2. 3. 56 J 1.

11 9 H 11 = ° ° ) + 5y + 2 ∠ m ∠D. Therefore,

. Find the given side length or side the given or LMNP. Find length

. 9 (9y + 17) ≅

. ¯ LM x ⋅ A

_ . BC Corresponding Parts a Proof in Corresponding Congruent ≅ (4x + 3) cm

6

= ( _ BC ∠ 8 - 13 = 35 cm , ¯

= GH

) ° ) K 8 G ( LM. △ACD Using Using + 2 ≅ quadrilateral △DEF = LMNP, ≅ Statements

≅ 6y - 13 → ( ≅ GH _ CD = GHJK - 13 = 6

≅ △ABD is the midpoint of of the midpoint D is △ABC ∠D GHJK _ each proof. Write △ABD ≅ △ACD BD is the midpoint of of D is the midpoint = 6x 2. 3. + 3 = 6x

1.

LM Therefore, Therefore, LM 4x Since Since Given: m∠D

DIFFERENTIATE INSTRUCTION DIFFERENTIATE Have students use geometry software to create designs using congruent triangles. congruent using designs create use to geometry software students Have positions, colors, different triangles using congruent multiple arrange They should congruent using one designs: separate three make them to Ask orientations. and using one isosceles triangles, and congruent using triangles, one equilateral scalenecongruent triangles. Technology Example 3

6. side. each 2 from Subtract side. each 5y from Subtract Module Module 3  Prove: Explain 3 Turn Your Quadrilateral So, m angle measure. Since Since  equation. an Write

GE_MNLESE385795_U1M03L3.indd 142

CorrectionKey=NL-A;CA-A

DO NOT EDIT--Changes must be made through "File info" "File through made be must EDIT--Changes NOT DO

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” therefore hasequalmeasure. each pairofcorresponding sidesiscongruent and same color, using adifferent color for each pair. highlight pairsalso of corresponding sides inthe for each pair of corresponding Students vertices. can same inthe vertices color and using adifferent color corresponding vertex by designating corresponding howcan see each vertex is mapped to its Focus onModeling esn 3 . Lesson 143 MP.4 measure? Explain. corresponding parts musthavecorresponding parts equalmeasure. maps afigure to acongruent figure, pairsof since there isasequence ofrigid motionsthat motions preserveanglemeasure andlength, , ∠A ≅G, ∠B ≅ ∠F, ∠E, corresponding sideshave thesamemeasure. PRACTICES INTEGRATE MATHEMATICAL QUESTIONING STRATEGIES SUMMARIZE THE LESSON SUMMARIZE THE QUESTIONING STRATEGIES ELABORATE When examining congruent figures, students parts haveparts measure? equal Why do pairs of corresponding congruent What are statements? six congruency Suppose you know that CBA ≅ EFG. of two congruent figures has equal Can you say that apair of corresponding sides corresponding angles have same the measure? Can you say two figures are congruent iftheir No. You mustalsodetermine that the Yes. thefigures If are congruent, then CB ¯

≅ F, EF , ¯

CA CA ¯ Since rigid

≅ G, EG , ¯

BA ¯

≅ ∠C ≅ FG FG ¯

CorrectionKey=NL-A;CA-A DO NOT EDIT--Changes mustbemadethrough "File info" GE_MNLESE385795_U1M03L3.indd 143

© Houghton Mifflin Harcourt Publishing Company have equal/not lengths.” equal Students work together to complete sentences. the and ____are corresponding, and measure _____degrees. Corresponding sides corresponding their because angles have/don’t have measures. equal Angles ___ For example: two (triangles/quadrilaterals/figures) “The are or are not congruent students with sentence stems to help attributes the describe them of figures. the ask to them explain why two figures are congruent or noncongruent. Provide Have students work inpairs. Provide each student with aprotractor and ruler, and Connect Vocabulary LANGUAGE SUPPORT 8. Write proof. each B Module 3 Module 9.

Your Turn

Prove: Given: Prove: Given: Given: Given: Prove: 3. 2. 1. 4. 3. 2. 1.

¯ ¯ ¯ ¯ 1. 4. 3. 2. ∠VST △ Quadrilateral ST CD AD AD Quadrilateral ∠J ∠K ∠J SVT AD △SVT Quadrilateral

ST AD

bisects bisects _ _

_ ∠J Quadrilateral NPQR; ≅ ≅

≅

≅ ≅ ≅ bisects ∠VSW. bisects GH CD

GH ¯ ¯ ¯

≅ ∠WST

≅ ≅ ≅ ≅ ∠P ∠K ∠P ∠P CD

△ GH _ _ ≅

∠J ∠VSW. SWT △SWT

Statements Statements ≅

JKLM ABCD ABCD Statements ∠K JKLM ≅ ≅ quadrilateral ≅ quadrilateral EFGH; ≅ quadrilateral quadrilateral

NPQR EFGH J 143 3. 2. 1.

3. 4. 2. 1. Definition ofanglebisector. figures are congruent. Corresponding ofcongruent parts Given Transitive ofCongruence Property figures are congruent. Corresponding of parts Given Given 3. 1. 2. 4. M

Transitive ofCongruence Property figures are congruent. Corresponding ofcongruent parts Given Given D A S L Reasons Reasons C K N Reasons B congruent H E R T G W Q V Lesson 3 Lesson F P 21/03/14 2:11PM 144 Exercises 1 Exercises 2–5, Exercises 10–13 6–9 Exercises 14–16 Exercises Practice Have students consider whether two consider students Have Concepts and Skills Concepts Explore of Parts of Exploring Congruence Figures Transformed Example 1 Parts of Congruent Corresponding Congruent are Figures Example 2 the Properties of Applying Congruence Example 3 Corresponding Congruent Using Parts in a Proof EVALUATE EVALUATE INTEGRATE MATHEMATICAL MATHEMATICAL INTEGRATE PRACTICES ASSIGNMENT GUIDE ASSIGNMENT DO NOT EDIT--Changes must be made through "File info" "File through made be must EDIT--Changes NOT DO CorrectionKey=NL-A;CA-A MP.1 Focus on Math Connections on Math Focus quadrilaterals, both with side lengths of 1 foot on on 1 foot of lengths both side with quadrilaterals, recognize should Students congruent. are each side, a rhombus. of that is the description that bottom and top open an with a box that Demonstrate the side although rigid, and not is side its on lying the pushed, is side when one the same stay lengths figures the two for possible is it Thus change. angles be not and measures angle different described have to congruent. 21/03/14 2:11 PM

© Houghton Mifflin Harcourt Publishing Company Lesson Lesson 3 ≅ ∠Z. ∠D Congruent are Figures of Congruent Parts Corresponding • Online Homework • Hints and Help • Extra Practice ∠G. G ≅ ∠Y, ≅ H ∠C 43° ∠D Z W XYZ must also be a right 43° ≅ ∠X, △ GHJ, = 42 ft Mathematical Practices Mathematical ≅ ∠B 25° Modeling Precision Reasoning Modeling Logic Reasoning DEF m∠G CORE △ = X COMMON have to be to a right have ≅ ∠W, J MP.4 MP.6 MP.2 MP.4 MP.3 MP.2 ∠A F

Y Since Since m∠D m∠D C WZ, ¯

3. B

≅ 144 , doesXYZ △ AD ¯

31 ft YZ, △ XYZ ¯

≅ ≅

.

112° ¯ A JH CD E

_ , ≅

XY ¯ D ¯

D = 31 ft. FE 19 ft

≅ XYZ triangle. is a right . How many additional congruence statements can you you can statements congruence additional many FGHJK. How

BC ¯

△ GHJ, WX, ¯ ≅

≅ Skills/Concepts Skills/Concepts Skills/Concepts Recall of Information Recall of Information Recall of Information Depth of Knowledge (D.O.K.) Depth of Knowledge 2 2 2 1 1 1 = 31 ft, so JH DEF is a right triangle, one of its angles is a right angle. Since corresponding corresponding Since angle. one of its angles is a right triangle, PQR is a right . What . What WXYZ quadrilateral of ABCD perfectly fit top on AB ¯

△ JH is a right triangle and △ PQR triangle and a right is = . Find the given side length or angle measure. the side given or △GHJ. Find length FE Since Since 1 . Therefore, those sides and angles are those sides and angles are Therefore, sides and angles of WXYZ corresponding . ABCD to 2–5 6–9 △PQR ≅ 14–16 19–22 17–18 10–13, Evaluate: Homework and Practice Homework Evaluate: There are five statements using the congruent corresponding sides and five statements statements sides and five corresponding using the congruent statements five are There angles. corresponding using the congruent Yes; since △ since Yes; angles of △ one of the congruent, are parts figures of congruent Yes; since the corresponding sides of congruent triangles are congruent, the sum of the the sum of congruent, triangles are sides of congruent the corresponding since Yes; both triangles. same for must be the sides (perimeter) lengths of the which means △ angle, Danielle finds that she can use a translation and a reflection to make make a reflection to and use a translation can she finds that Danielle quadrilateral Essential Question Essential Check-In ABCDE pentagon that know you Suppose pentagon to congruent is of angles and the sides using write Danielle can statements congruence Why? the quadrilaterals? triangle? Why or why not? why or triangle? Why Explain. the pentagons? parts of corresponding using write JH If A student claims that any two congruent triangles must have the same perimeter. perimeter. the same have triangles must congruent two any that claims A student Explain. agree? Do you The same sequence of rigid motions that maps ABCD to maps WXYZ of rigid motions that same sequence also maps sides and angles The of congruent: Exercise

1. 12. Module Module 3 2. △DEF 11. 10. Elaborate Elaborate

GE_MNLESE385795_U1M03L3.indd 144

CorrectionKey=NL-A;CA-A

DO NOT EDIT--Changes must be made through "File info" "File through made be must EDIT--Changes NOT DO

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 . Lesson 145 explain why it is not correct. statement that is not correct for diagram the and diagram. Then have write them acongruence other correct congruence statements for same the statements for agiven diagram, and ask to them write Focus onCommunication MP.3 PRACTICES INTEGRATE MATHEMATICAL Have students compare congruence their CorrectionKey=NL-C;CA-C DO NOT EDIT--Changes mustbemadethrough "File info" GE_MNLESE385795_U1M03L3.indd 145

Exercise © Houghton Mifflin Harcourt Publishing Company 24–25 23 27 26 △STU DEFG △ 4. KLMN Module 3 Module 12. 10. △GHJ ≅ 6. 8. ABC ≅ Cong., andcorr. of≅fig. ≅. parts △ m∠R GJ GH Congruent segments have thesamelength. of Cong., andcorr. of≅fig. ≅. parts △ m∠M _

30 BC FG 6x So, 2y 32 So, So, . Justify your answers. ≅ GHJ GHJ ≅ =

≅ + 2=5x + 2=32cm. + 3=4y + 3=35in. FG . Find the given side lengthKLMN. Find or given the side measure. angle

¯ BC

BC PQRS length. Find or given the side measure. angle

△PQR ¯ SU 3 3 3 2 Depth ofKnowledge (D.O.K.) ∠M FG △ =

ST

¯ Strategic Thinking Strategic Thinking Strategic Thinking Skills/Concepts ≅ ≅

. Find the given side lengthTUV. Find or given the side measure. angle = 2y

= 6x

≅ m∠R ≅

△ △ C ≅ 2.1 cm

UV ¯ ¯ MN. G STU by the Transitive Prop. of STU by the Transitive Prop. and △PQR and ∠R. + 3=2 + 7→ N + 2=6 . - 29→ (6x +2)cm

= D So,

K So, 75° 79°. (20x +12)° (3y +2)°

BC

FG FG (2y +3)in. x ( 2.9 cm ( = UV. A 16 5 = 5 = MN. 16 ≅ )

) + 2= =

+ 3= △STU B y 79° L (6x -1)cm

E . Complete following using the or of aside angle

M F V N (5x +7)cm 68° K R (25x -8)° 145 (4y -29)in. (y +9)in. 5. T 7. 9. MP.3 MP.6 MP.3 MP.2 COMMON 13. 11. Q CORE PS 2.1 cm (4y)° m∠U m∠D So, m So,

3y (

( 20x So, m So, Logic Precision Logic Reasoning 4 U 20 angles have thesamemeasure. Cong., andcorr. of≅fig. ≅. Cong. parts △ ∠J m (4y -18)°

Cong., andcorr. of≅fig. ≅. parts △ ⋅ 20-18 ¯ + 2=4y KN ∠G GHJ S ⋅ 4+12 ≅ + 12=25x GHJ

∠U Mathematical Practices ∠B ∠D L ∠D

≅

= ∠U ≅ M

¯ PS.

= ≅ m∠S ≅

= P ≅ △

( △

∠U. )

(

) ∠K. - 18→ KN 4y STU by the Transitive Prop. of ° 20x °=92°. STU by the Transitive Prop. of = 62°. - 18 = So, So, - 8→4= + 12 PS m∠B m∠D ) = 20 ° ) = °

= =

= = y

m∠U.

x m∠K. Lesson 3 Lesson 6/9/15 12:23AM 146 AVOID COMMON ERRORS COMMON AVOID Students may find the value of a variable or the value the value or a variable of find the value may Students a to the solution as expression algebraic an of the way part of fact in only when they are problem to students Remind process. the solving through sure make to the question initial to back go always the problem. to the solution is the answer 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 5/14/14 6:59 PM

© Houghton Mifflin Harcourt Publishing Company Lesson 3 R Congruent are Figures of Congruent Parts Corresponding S bisector ≅ ≅ ≅ ≅ C Q Reasons Reasons Reasons fig. are are fig. fig. are are fig. are fig. fig. are are fig. ≅ ≅ ≅ ≅ B D T P Given Definition of midpoint Definition of segment Corr. parts of Corr. Given Given Property of Congruence Transitive Given Definition of angle bisector Corr. parts of Corr. parts of Corr. Corr. parts of Corr.

A 4. 3. 2. 1. 4. 2. 3. 1. 2. 3. 1. 4. F U K ∠E ≅ 146 ∠D QRST G ∠BCD. FGHJK J H

bisects bisects ∠BCD.

C D quadrilateral quadrilateral _ AC ¯ AC PR. quadrilateral QRST≅ quadrilateral ¯

≅ pentagon pentagon B ≅ pentagon FGHJK; ≅ and and Statements . Statements Statements

PQTU

_ PR △ ADC

BAD ∠BAD ¯ PR. ABCDE

≅ △DEF ∠K ∠DCA E A ∠DAC ≅ ≅ ≅ ≅ ∠E ∠K ¯ ∠K

QR bisects

bisects ∠BAD

≅ ≅ ≅

bisects bisects ≅ _ ∠D

QT △ ABC _

AC PQ AC QT is the midpoint of Q is the midpoint Quadrilateral Quadrilateral ∠D ∠E △ABC ∠BAC ∠BCA ∠D Pentagon Pentagon ¯ ¯

¯

3. 1. 2. 4. 3. 4. 4. 1. 1. 2. 3. 2. Prove: Prove: Prove: Prove: Given: ABCDE Pentagon Given: PQTU Quadrilateral Given: Prove:

16. Module 3 15. 14. Write each proof. Write

GE_MNLESE385795_U1M03L3 146

CorrectionKey=NL-B;CA-B

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 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 . Lesson 147 positions yield pairs of corresponding sides. similar way, pairs of letters that are incorresponding respective This triangles. means ∠J≅M.In a appearboth first position inthe names inthe of their For example, in JZQ ≅ MDH corresponding positions inacongruence statement. corresponding angles by choosing pairs of letters in shouldThey know that can identify they inacongruencevertices statement is not random. Make sure understand they that order the of the Students may write incorrect congruence statements. AVOID COMMON ERRORS , the letters, the M Jand CorrectionKey=NL-C;CA-C DO NOT EDIT--Changes mustbemadethrough "File info" GE_MNLESE385795_U1M03L3.indd 147

© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Ken Brown/E+/Getty Images Module 3 Module 19. 18. 17. △ABC ≅ 20. b. The figure shows dimensions the of two city parks, where △ RST m m in which shows of part horizontal the of beam atower crane building materials at construction sites. The figure A tower crane tosteel, lift is used concrete, and What is total the length of fences the required to surround parks? the and and a. Since 210 Since thetrianglesare congruent, they have thesameperimeter, whichis A

∠C ∠D A member of construction the crew claims that agree? Explain. Is it possible to determine m Yes; since corr. are ≅, parts m∠GBH This meansAC Yes; since corr. are ≅,m∠ABG parts YX

_ + 320 59° △DEF △RST G Since m∠EFM Since m∠JAB

≅ ≅ △ABG YZ

_ . Find the given side length. Find or given the side measure. angle . Acity employee wants to order fences new to surround parks. both = ≅

△ABC △ABC J A

S + 180° , △XYZ + 62 BAC m∠BAC m∠EFD △BCH 850 ft. = 850ft. The total lengthofthefences is850+=1700ft. ° istwice AB. ≅ ≅ B -59°

△DEF, △DEF, ¯ ST H

320 ≅ = ≅

= -27°

R

f △HGB ¯ BAC 90°, so62°+m∠BAC YZ,

t 180°, so71°+m∠EFD ∠C BAC ∠BAC ∠GBH C 27° soST

B =

AB _ ≅

T 94°. ∠EFD.

≅ ≅ ? If so, how? If not, why not? M = ∠D. C

BC _

210

Z YZ 71

andsoBisthemidpoint of = E BAC m∠BAC

f ° m∠C

t 320 ft. Since= 320ft. F 27° andm∠HBC 147 ,

X

AC _ = is twice as islong twice as =

m∠EFD, and = = BAC 90° andm∠BAC BAC m∠D, andBAC 180° andm∠EFD

¯

Y ≅ YX YX D =

△ XYZ ≅

59°, so YZ, ¯ AB _

YX YX =

. Do you. Do AC _ = YZ 28°. = = = .

109°. 109°, som∠C 28°, som∠D = 320ft. = = Lesson 3 Lesson 28°. 109°. 6/9/15 12:23AM 148 When students solve algebraic equations to to equations algebraic solve students When INTEGRATE MATHEMATICAL MATHEMATICAL INTEGRATE PRACTICES DISCUSSION PEERTOPEER info" "File through made be must EDIT--Changes NOT DO CorrectionKey=NL-A;CA-A MP.2 to discuss a partner with to how students Ask Have congruent. are figures whether two determine the look for figures, of a pair other each give students a then write parts, and corresponding congruent the Repeat the figures. for statement congruence figures. of pairs other for exercise Focus on Reasoning Focus find the measures of congruent corresponding parts corresponding congruent of find the measures the verify first them to that caution figures, of students that Suggest correct. are correspondences parts. corresponding of the pairs listing start by 05/04/14 4:19 AM © Houghton Mifflin Harcourt Publishing Company• Image Credits: ©(T)Image Collective/Alamy, (B) ©Oleksiy Maksymenko Photography/Alamy

QR, QR, Lesson 3 QS, ¯ ¯ ¯

≅ ∠Q

≅

≅ ≅ ≅ ∠S, ≈ 9.3 s. t has two has two MP MP MN ∠M = 52 and ¯ ∠P ¯ ¯

Congruent are Figures of Congruent Parts Corresponding

→ 71°. ° = ) △QRS, △QRS, △QRS, △QRS, △QRS, + 19. Solving the x ≅ ≅ ≅ ≅ ° = ≅ ) - 33 33 = 40 ft. 2x (

52 mm. Therefore, Therefore, = 52 mm. 52 mm. The perimeter of perimeter The = 52 mm. = m∠N. = 52 mm. △QRS MN. △MNP △MNP △MNP △MNP △MNP = = QR MP QR ⋅ 52 -

so CD 2 Since Since so △QRS is 52 + 52 + 34 = 138 mm. so MP Since Since so sides with the same length, so it is isosceles. Since Since Since Since since corresponding parts corresponding of since congruent. are figures congruent

( Since Since so 2x - 33 = equation shows that x that shows equation m∠P

CD,

¯

= 5(6) + 1 = 31 ft P d. c. b. e. a.

≅ and so m∠K AD ¯

∠N ≅ False False False False False

148 (x + 19)° △CBD. S ≅ 34 mm (15x + 15)° R C N

in the figure. △MNP in the figure. = 4(6) - 4 = 20 ft,KL ≅ True True True True True

so AB = 50 ft. Also, 50 ft M

L CB, ¯

52 mm = 6; JK

≅ x B D .

AB ¯

Q 30 ft ¯ MN (5x + 1) ft

M 41 ft + 15 → 40 ft . Determine whether each statement whether. Determine statement each △CBD, = 50 + 50 + 40 = 140 ft; = rate so 140= 15t distance × time, (20x - 15)° ≅ (2x - 33)° A company installs triangular pools at hotels. All hotels. pools installs triangular at A company CD K A

∠Q △QRS P is isosceles. 120 mm. is + = 52° ≅ ≅ is longer than than longer is

△ABD BC J - 15 = 15x _ MP + △QRS m∠P ∠M The perimeter of perimeter of The △QRS 52 mm

AB Since Since Since corresponding parts are congruent, ∠K parts congruent, corresponding are Since (4x - 4) ft △MNP is also 92 ft. of perimeter The of △JKL is 20 + 31 + 41 = 92 ft. perimeter The 20x N a. △MNP about the true triangles false. is Select or about the correct part. lettered each for answer radio-controlled trucks. In the figure, △ABD the figure, trucks. In radio-controlled per speed 15 feet second. of a constant at trucksThe travel from the course on take a truck travel does to it long How a second. of tenth A to the nearest B to to C to D? Round Kendall and Ava lay out the course shown below for their for below shown the course out lay Ava and Kendall What is the perimeter of each pool? each the perimeter is of What of the pools are congruent and △JKL and congruent the pools are of Multi-Step b. c. d. e.

23. 22. 21. Module 3

GE_MNLESE385795_U1M03L3.indd 148

CorrectionKey=NL-A;CA-A

info" "File through made be must EDIT--Changes NOT DO

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 . Lesson 149 figures oflabeled as part entry. journal the words. Encourage to them include one or more Congruent Figures Are Congruent own intheir statement the discuss that Corresponding Parts of Have students inwhich they write entry ajournal JOURNAL CorrectionKey=NL-A;CA-A DO NOT EDIT--Changes mustbemadethrough "File info" GE_MNLESE385795_U1M03L3.indd 149

© Houghton Mifflin Harcourt Publishing Company 25. 24. Module 3 Module 26. 27. H.O.T. is not known and cannot bedetermined andcannot is notknown from thegiven information. No; thesideof△ABCthat corresponds to corresponding angleto becongruent. ∠F,sothetrianglescannot congruent. Since m∠F No; ifthetriangleswere congruent, thencorresponding angleswould be correct answer. correct work is shown below. Explain error the and the find △GHJ prove If this? so, write proof. the If not, explain why not. not, explain why not. m∠E said that point R Explain theError Justify Reasoning Critical Thinking Analyze Relationships possible to determine length the of 5. 4. 3. 2. 1.

Student's Work

Focus onHigherOrder Thinking ¯ ¯ ¯ R isthemidpoint of △PQR RP RS RP = 50°,and m∠F ≅ GH ≅ 5x ≅ ≅

△RST

¯ ¯ ¯ RT RT RS RT RT - 2=6x Statements -2 = 5x ≅ 3

△SQR = = and was asked The toGH. student’s find appears midpoint the to be of - 2=5 x x

- 5 In In

A student was told that - 5 ABC △ABC that Given , m∠A △ABC = 65°.Is it possible for congruent? to triangles the be Explain.

( = 65°,there isnoangleof△ABCthat could bethe 3 △PQR

¯ PT PT )

- 2=13m

≅ EF _ 1. 5. 4. 3. 2. △SQR = 55°,m∠B

? If so, length the and find your justify steps. If

Definition ofmidpoint Transitive Property Given of ≅ figs Corr parts Given ≅ △DEF and and PT

_ ¯ EF Reasons 5x sides. Since △GHJ The student incorrectly identified corresponding RS

_ 149 , . Is it possible to = 50°,and m∠C

AB

is

- 2=4x ≅

¯ BC RT

2.7 ft, and= 2.7ft, AC _ Yes; . The lengthofthisside . A student A .

G . are ≅ + 3→ (5x -2)m = 75°.In △DEF J ≅

x 3.4 ft, is it= 3.4ft, Q △RST, = 5;GH H

¯ GH S = 5(5)-223m. (4x +3)m ≅ ,

¯ RS. T S P

(6x -5)m R R Lesson 3 Lesson T 21/03/14 2:15PM 150 3 4 2 5 6 1 8 7 Sample answer: Rotate Triangle 1 Triangle Rotate Sample answer: Given Triangle 1, how could you find the you could 1, how Triangle Given seven triangles using the other of locations Describe how, starting with a square, you you a square, with starting Describe how, Sketch and number the eight inner triangles inner the eight number and Sketch Sample answer: Draw the diagonals of the the Draw Sample answer: INTEGRATE MATHEMATICAL MATHEMATICAL INTEGRATE PRACTICES INTEGRATE MATHEMATICAL MATHEMATICAL INTEGRATE PRACTICES transformations? transformations? MP.3 MP.8 Focus on Communication Focus Focus on Patterns Focus could draw the pattern of a Yankee Puzzle Puzzle a Yankee of the pattern draw could quilt. of the Yankee Puzzle quilt on the board. on quilt Puzzle the Yankee of square. Find the midpoints of the four sides. sides. of the four the midpoints Find square. of each side with the the midpoint Connect it and the midpoint to side adjacent of the midpoint it. of the side opposite clockwise around the center the center 90°, 180°, and 270° clockwise around triangles 3, 5, and 7. Reflect locate to point locate line to the vertical 1 across center Triangle 2 90°, 180°, and Triangle rotate Then 2. Triangle locate to point the center 270° clockwise around triangles 4, 6, and 8. 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: 21/03/14 2:10 PM © Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Ken Brown/E+/Getty Images Lesson Lesson 3 Congruent are Figures of Congruent Parts Corresponding

to the position to AB ¯

150 D

.

can be transformed to to be can transformed . _ CD _ AB AB

= . One way is to translate it to the position of the it to translate is to One way CD. ¯ C

B A because corresponding parts of congruent figures are congruent. are parts figures AB because corresponding of congruent = quilt. quilt. design consists of 4 of the triangles joined to form a square. a square. form of 4 of the triangles joined to design consists the position of the triangle with base the base triangle with of the position CD counterclockwise about C, about it 90° counterclockwise then, rotate triangle directly it, beneath the right. to then translate of the triangle with base of the triangle with base The design should be square. design should The only. quadrilaterals triangles and/or of consist design should The symmetry. rotational 90-degree have design should The Explain how the triangle with base the triangle base with how Explain CD that know you how Explain c. b. a. describe to the the design of shapes congruent the idea of Use • • • The illustration shows a “Yankee Puzzle” quilt. quilt. Puzzle” a “Yankee shows illustration The

EXTENSION ACTIVITY EXTENSION Challenge students to draw and color a design for a quilt that meets the following meets the following that a quilt for a design color and draw to students Challenge requirements:

Module Module 3 with base the triangle transform to ways many are There quarter Each triangles. of the 16 congruent from design is created The Lesson Lesson Task Performance

GE_MNLESE385795_U1M03L3.indd 150

CorrectionKey=NL-A;CA-A

DO NOT EDIT--Changes must be made through "File info" "File through made be must EDIT--Changes NOT DO

CorrectionKey=NL-C;CA-C DO NOT EDIT--Changes must be made through “File info” “File through be made must EDIT--Changes DO NOT