Essential Question: How can you be sure that the result of a construction is valid? Explore 1 Using a Reflective Device to Construct

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1 OMMON ORE n Locker LESSON Justifying onstructions oon ore Math Standards The student is expected to: OMMON ORE G-O..12 Make fora geoetric constructions with a variety of toos and ethods (copass and straightedge, string, refective devices, paper foding, dynaic geoetric software, etc.). so G-O..13, G-SRT..5 Matheatica ractices OMMON ORE 6.1 M.3 Logic Language Objective Expain in your own words how to construct a copy of an ange. Nae ass ate 6.1 Justifying onstructions Essentia uestion: How can you be sure that the resut of a construction is vaid? Expore 1 Using a Refective evice to onstruct a erpendicuar Line You have constructed a ine perpendicuar to a given ine through a point not on the ine using a copass and straightedge. You can aso use a refective device to construct perpendicuar ines. Step 1 ace the refective device aong ine. Look through the device to ocate the iage of point on the opposite side of ine. raw the iage of point and abe it '. Step 2 Use a straightedge to draw '. Resource Locker ENGGE Essentia uestion: How can you be sure that the resut of a construction is vaid? You can use geoetric properties to prove that the drawn figure ust have the properties of the expected shape. REVIEW: LESSON ERFORMNE TSK View the Engage section onine. Make sure students understand the eaning of toerance in the context of anufacturing. Then preview the Lesson erforance Task. Houghton Miffin Harcourt ubishing opany Expain why ' is perpendicuar to ine. y the definition of refection, ine is the perpendicuar bisector of '. ace the refective device so that it passes through point and is approxiatey perpendicuar to ine. djust the ange of the device unti the iage of ine coincides with ine. raw a ine aong the refective device and abe it ine n. Expain why ine n is perpendicuar to ine. n The refective device is positioned so that for any point on ine, the iage,, is aso on ine. Since ine n is the ine of refection, it ust be perpendicuar to the ine through and, which is ine. Modue Lesson 1 Nae ass ate 6.1 Justifying onstructions Essentia uestion: How can you be sure that the resut of a construction is vaid? G-O..12 Make fora geoetric constructions with a variety of toos and ethods (copass and straightedge, string, refective devices, paper foding, dynaic geoetric software, etc.). so G-O..13, G-SRT..5 Houghton Miffin Harcourt ubishing opany Expore 1 Using a Refective evice to onstruct a erpendicuar Line You have constructed a ine perpendicuar to a given ine through a point not on the ine using a copass and straightedge. You can aso use a refective device to construct perpendicuar ines. Step 1 ace the refective device aong ine. Look through the device to ocate the iage of point on the opposite side of ine. raw the iage of point and abe it '. Step 2 Use a straightedge to draw '. Expain why ' is perpendicuar to ine. y the definition of refection, ine is the perpendicuar bisector of '. ace the refective device so that it passes through point and is approxiatey perpendicuar to ine. djust the ange of the device unti the iage of ine coincides with ine. raw a ine aong the refective device and abe it ine n. Expain why ine n is perpendicuar to ine. Resource HROVER GES Turn to these pages to find this esson in the hardcover student edition. The refective device is positioned so that for any point on ine, the iage,, is aso on ine. Since ine n is the ine of refection, it ust be perpendicuar to the ine through and, which is ine. Modue Lesson Lesson 6.1

2 Refect 1. How can you check that the ines you drew are perpendicuar to ines and? Use a protractor to easure the ange fored by ' and ine, and the ange fored by 2. Use the refective device to draw two points on ine that are refections of each other. Labe the points X and X'. What is true about X and X'? Why? Use a ruer to check your prediction. X = X. ' is the perpendicuar bisector of XX ', and any point on the perpendicuar 3. escribe how to construct a perpendicuar bisector of a ine segent using paper foding. Use a rigid otion to expain why the resut is a perpendicuar bisector. Fod the paper so that one endpoint of the ine is apped to the other endpoint. The fod is a refection, and the crease is the ine of refection. y the definition of refection, the crease is the perpendicuar bisector of the segent that connects a point and its iage. Expore 2 Justifying the opy of an nge onstruction You have seen how to construct a copy of an ange, but how do you know that the copy ust be congruent to the origina? Reca that to construct a copy of an ange, you use these steps. ine n and ine. In both cases, the ange shoud easure 90. bisector of a segent is equidistant fro the endpoints of the segent. Step 1 raw a ray with endpoint. Step 2 raw an arc that intersects both rays of. Labe the intersections and. Step 3 raw the sae arc on the ray. Labe the point of intersection E. Step 4 Set the copass to the ength. Step 5 ace the copass at E and draw a new arc. Labe the intersection of the new arc F. raw F. is congruent to. Sketch and nae the two trianges that are created when you construct a copy of an ange. F F E Houghton Miffin Harcourt ubishing opany EXLORE 1 Using a Refective evice to onstruct a erpendicuar Line INTEGRTE TEHNOLOGY Students ay use geoetry software to ode the construction of a perpendicuar to a given ine. UESTIONING STRTEGIES What ange easure coud you use to describe a straight ine? What ange easures does the construction of a perpendicuar ine produce? escribe the reationship between the ange easure of the straight ine and the anges created by the perpendicuar ine. 180 ; 90 ; the perpendicuar divides the 180 ange into two equa haves of 90 each. EXLORE 2 Justifying the opy of an nge onstruction INTEGRTE TEHNOLOGY Students ay use geoetry software to ode the construction of an ange congruent to a given ange. E Modue Lesson 1 ROFESSIONL EVELOMENT Math ackground opass and straightedge constructions date to ancient Greece. In fact, one of the cassic probes of ancient Greek atheatics was the trisection of the ange. That is, using a copass and straightedge, is it possibe to construct an ange whose easure is one-third that of an arbitrary given ange? It was not unti 1837 that this construction was proven to be ipossibe. On the other hand, it is a straightforward task to bisect any ange, and students prove the vaidity of this fundaenta construction in this esson. Justifying onstructions 274

3 UESTIONING STRTEGIES How do you know that the sides of the trianges are congruent? They were constructed using the sae copass setting. EXLIN 1 roving the nge isector and erpendicuar isector onstructions What segents do you know are congruent? Expain how you know. E because the arcs that created these segents have the sae radius. EF because the arcs that created these segents have the sae radius. F because the arcs that created these segents have the sae radius. re the trianges congruent? How do you know? Yes. They are congruent by the SSS Triange ongruence Theore. Refect 4. iscussion Suppose you used a arger copass setting to create _ than another student when copying the sae ange. Wi your copied anges be congruent? Yes. Even though our trianges and EF wi not be congruent, our anges and wi be congruent because of the transitive property of congruence. VOI OMMON ERRORS Soe students ay use the anges resuting fro the bisected ange to prove the trianges are congruent by SS. Reind these students that they are trying to prove the congruence of those two anges, so they cannot be used in the justification. UESTIONING STRTEGIES What roe does the ange bisector pay in proving the two trianges are congruent? The ange bisector fors a side that is coon to both trianges, so it is congruent to itsef. OMMUNITE MTH Have students work in pairs to write the steps for constructing an ange bisector. The first student does the construction and expains the steps. The second student writes down the steps. The students switch roes and repeat the procedure with parae ine construction. Houghton Miffin Harcourt ubishing opany 5. oes the justification above for constructing a copy of an ange work for obtuse anges? Yes. The construction ethod is the sae, and it wi sti resut in two trianges with three pairs of congruent corresponding sides. So the two trianges are congruent, and the copy of the ange wi be congruent to the origina. Expain 1 roving the nge isector and erpendicuar isector onstructions You have constructed ange bisectors and perpendicuar bisectors. You now have the toos you need to prove that these copass and straightedge constructions resut in the intended figures. Exape 1 rove two bisector constructions. You have used the foowing steps to construct an ange bisector. Step 1 raw an arc intersecting the sides of the ange. Labe the intersections and. Step 2 raw intersecting arcs fro and. Labe the intersection of the arcs as. Step 3 Use a straightedge to draw _. rove that the construction resuts in the ange bisector. The construction resuts in the trianges and. ecause _ the sae _ copass setting was used to create the, and. The segent _ is congruent to itsef by the Refexive roperty of ongruence. So, by the SSS Triange ongruence Theore,. orresponding parts of congruent figures are congruent, so. y the definition of ange bisector, is the ange bisector of. Modue Lesson Lesson 6.1 OLLORTIVE LERNING eer-to-eer ctivity Have students work in pairs and give the the foowing propt: You know how to draw a perpendicuar bisector, copy a segent, and copy an ange. How can you use a sequence of these constructions to construct a square? Sape answer: You can draw a segent and construct its perpendicuar bisector to get a right ange. opy one of the haves of the origina segent on the perpendicuar bisector so that you now have two sides of the square. Then copy the right ange and a side ength two ore ties unti you have four right anges and four congruent sides.

4 You have used the foowing steps to construct a perpendicuar bisector. Step 1 raw an arc centered at. Step 2 raw an arc with the sae diaeter centered at. Labe the intersections and. Step 3 raw _. rove that the construction resuts in the perpendicuar bisector. The point is equidistant fro the endpoints of, so by the onverse of the erpendicuar isector Theore, it ies on the perpendicuar bisector of _. The point is aso equidistant fro the endpoints of, so it aso ies on the perpendicuar bisector of _. Two points deterine a ine, so is the perpendicuar bisector of. Refect 6. In art, what can you concude about the easures of the anges ade by the intersection of _ and _? The four anges are congruent 90º anges. 7. iscussion _ cassate cais that in the construction shown in art, is the perpendicuar bisector of _. Is this true? Justify your answer. The cai is true for the sae reason that is the perpendicuar bisector of. Your Turn 8. The construction in art is aso used to construct the idpoint R of MN _. How is the proof of this construction different fro the proof of the perpendicuar bisector construction in art? You need to add an extra step to say that because is the perpedicuar bisector of MN, the point of intersection wi be the idpoint of MN by the definition of idpoint. 9. How coud you cobine the constructions in Exape 1 to construct a 45 ange? First construct a perpendicuar bisector of a segent. This creates 90 anges. hoose one of the anges to bisect. This wi construct two 45 anges. M N R Houghton Miffin Harcourt ubishing opany Modue Lesson 1 IFFERENTITE INSTRUTION ritica Thinking Have students use a protractor to draw a 60 ange. Then chaenge the to use copass and straightedge constructions to create anges with the foowing easures: fter they copete as any as they can, have the use a protractor to check for accuracy. Justifying onstructions 276

5 VOI OMMON ERRORS When proving that a construction resuted in a perpendicuar bisector, soe students ay use the erpendicuar isector Theore. Make sure they understand why it is the onverse of the erpendicuar isector Theore that is used. ELORTE ONNET VOULRY oint out to students that the word converse coes fro the Latin converses, which eans to turn around. Eaborate 10. escribe how you can construct a ine that is parae to a given ine using the construction of a perpendicuar to a ine. onstruct a perpendicuar to the given ine. Then construct a perpendicuar to the perpendicuar. This ine is parae to the given ine. 11. Use a straightedge and a piece of string to construct an equiatera triange that has as one of its sides. Then expain how you know your construction works. (Hint: onsider an arc centered at with radius and an arc centered at with radius.) _ since and are radii of the sae circe., since and are radii of the sae circe. So, the three sides of the triange a have the sae ength. 12. Essentia uestion heck-in Is a construction soething that ust be proven? Expain. ossibe answer: Yes. To know that a construction is vaid, you ust show that the resuting figure has the properties that are described by the nae of the construction. For exape, for the construction of a congruent ange, you ust prove the anges are congruent. SUMMRIZE THE LESSON Why is it iportant to justify and prove constructions? The construction itsef is just a concrete exape. roving the construction ensures that it works for every possibe exape. Houghton Miffin Harcourt ubishing opany Evauate: Hoework and ractice 1. Juia is given a ine and a point not on ine. She is asked to use a refective device to construct a ine through that is perpendicuar to ine. She paces the device as shown in the figure. What shoud she do next to draw the required ine? Onine Hoework Hints and Hep Extra ractice She shoud adjust the ange of the refective device unti the iage of ine coincides with ine. Then she shoud draw a ine aong the edge of the refective device. This is the required ine. 2. escribe how to construct a copy of a segent. Expain how you know that the segents are congruent. raw a segent onger than the segent to be copied. Set the copass opening to the ength of the given segent. Using that copass setting, draw an arc centered at one endpoint of the segent you drew. Mark the intersection to abe the copied segent. The segents are congruent because the copass ensures the two segents have the sae ength. Modue Lesson 1 LNGUGE SUORT onnect Vocabuary nayze the parts of the word bisect or bisector with students. They previousy identified the prefix bi- as eaning two. The root sect coes fro the word secare, which eans to cut. So, to bisect eans to cut into two equa parts. 277 Lesson 6.1

6 opete the proof of the construction of a segent bisector. 3. Given: the construction of the segent bisector of _ rove: bisects _ EVLUTE Stateents 1. = and =. 2. is on the perpendicuar bisector of _ is on the perpendicuar bisector of _. 3. Reasons 1. Sae copass setting used 4. is the perpendicuar bisector of _. 4. Through any two points, there is exacty one ine. 5. bisects _. 5. efinition of perpendicuar bisector 4. opete the proof of the construction of a congruent ange. Given: the construction of given HFG rove: HFG onverse of the erpendicuar isector Theore onverse of the erpendicuar isector Theore SSIGNMENT GUIE oncepts and Skis Expore 1 Using a Refective evice to onstruct a erpendicuar Line Expore 2 Justifying the opy of an nge onstruction Exape 1 roving the nge isector and erpendicuar isector onstructions ractice Exercise 1 Exercises 2 15 Exercises n H F G Stateents Reasons 1. FG = FH = = 1. sae copass setting 2. GH = 2. Sae copass setting used 3. FGH SSS Triange ongruence Theore HFG 4. T Houghton Miffin Harcourt ubishing opany Modue Lesson 1 Exercise epth of Knowedge (.O.K.) OMMON ORE Matheatica ractices 1 2 Skis/oncepts M.5 Using Toos Skis/oncepts M.6 recision 19 3 Strategic Thinking M.1 robe Soving 20 3 Strategic Thinking M.6 recision Justifying onstructions 278

7 INTEGRTE MTHEMTIL RTIES Focus on ounication M.3 Mode precise atheatica vocabuary when discussing the constructions in the exercises. Using ters such as intersect, arc, and congruent, and naing figures by their correct geoetric naes hep accusto students to counicate using atheatica anguage. To construct a ine through the given point, parae to ine, you use the foowing steps. Step 1 hoose a point on ine and draw _. Step 2 onstruct an ange congruent to at. Step 3 onstruct the ine through the given point, parae to the ine shown. escribe the reationship between the given anges or segents. Justify your answer. 5. TS and UR They are congruent; 6. SU and RU They are suppeentary; these are the origina ange and the sae-side interior anges of parae ines ange that was copied. are suppeentary. T V W U S 1 R 7. VU and UR They are congruent; 8. TS and WU They are suppeentary; aternate interior anges of parae WU and UR for a inear pair and ines are congruent. are suppeentary, and TS UR, so TS and WU are suppeentary. 9. _ U and S _ They are congruent; 10. radii of congruent circes are congruent. _ U and _ T They are congruent; radii of congruent circes are congruent. Houghton Miffin Harcourt ubishing opany 11. To construct a ine through the given point, parae to ine, you use the foowing steps. Step 1 raw ine through and intersecting ine. Step 2 onstruct an ange congruent to at. Step 3 onstruct the ine through the given point, parae to the ine shown. How do you know that ines and n are parae? Expain. Line is a transversa of ines and n. nges 1 and 2 are congruent corresponding anges, because ange 2 is a constructed copy of ange 1. y the onverse of the orresponding nges ostuate, n. 12. onstruct an ange whose easure is 1 the easure of Z. Justify the 4 construction. isect Z and then bisect one of the saer anges. 1_ 2 ( 1_ 2 Z ) = 1_ 4 Z. Z 1 2 F n Modue Lesson Lesson 6.1

8 In Exercises 13 and 14, use the diagra shown. The diagra shows the resut of constructing a copy of an ange adjacent to one of the rays of the origina ange. ssue the pattern continues. 13. If it takes 10 ore copies of the ange for the ast ange to overap the first ray (the horizonta ray), what is the easure of each ange? There wi be tota of 12 copies of the ange. The su of the easures wi be 360. _ 360 = 30; The easure of each ange is If it takes 8 ore copies of the ange for the ast ange to overap the first ray (the horizonta ray), what is the easure of each ange? There wi be tota of 10 copies of the ange. The su of the easures wi be 360. _ 360 = 30; The easure of each ange is Sonia draws a segent on a piece of paper. She wants to find three points that are equidistant fro the endpoints of the segent. Expain how she can use paper foding to hep her ocate the three points. Fod the paper so that the segent s endpoints coincide. The fod ine is the segent s perpendicuar bisector, so any three points on the fod ine wi be equidistant fro the endpoints. OGNITIVE STRTEGIES Have students think about the steps used with each construction. sk the to refect on how they reeber how to do each one. In sa groups, have students discuss strategies for reebering the steps. Then share each group s best strategies with the cass. In Exercises 16 18, a poygon is inscribed in a circe if a of the poygon s vertices ie on the circe. 16. Foow the given steps to construct a square inscribed in a circe. Use your copass to draw a circe. Mark the center. raw a diaeter, _, using a straightedge. onstruct the perpendicuar bisector of _. Labe the points where the perpendicuar bisector intersects the circe as and. Use the straightedge to draw _, _, _, and _. 17. Suppose you are given a piece of tracing paper with a circe on it and you do not have a copass. How can you use paper foding to inscribe a square in the circe? Fod the circe so that one haf coincides with the other. The crease is a diaeter. Then fod the diaeter onto itsef to ake another crease that is the perpendicuar bisector of the diaeter. The two creases deterine the four vertices of the square on the circe. Houghton Miffin Harcourt ubishing opany Modue Lesson 1 Justifying onstructions 280

9 VOI OMMON ERRORS Soe students ay use a ruer or protractor to ake easureents during their constructions. Reinforce that easuring toos shoud be used ony after a construction to verify the resuts. Measureents shoud not be ade during a construction. JOURNL 18. Foow the given steps to construct a reguar hexagon inscribed in a circe. Tie a penci to one end of the string. Mark a point O on your paper. ace the string on point O and hod it down with your finger. u the string taut and draw a circe. Mark and abe a point. Hod the point on the string that you paced on point O, and ove it to point. u the string taut and draw an arc that intersects the circe. Labe the point as. Hod the point on the string that you paced on point, and ove it to point. raw an arc to ocate point on the circe. Repeat to ocate points, E, and F. Use your straightedge to draw EF. E F O Have students expain the iportance of proving constructions. H.O.T. Focus on Higher Order Thinking 19. Your teacher constructed the figure shown. It shows the construction of ine T through point and parae to ine. a. opass settings of ength and were used in the construction. opete the stateents: T With the copass set to ength, an arc was drawn with the copass point at point. With the copass set to ength, an arc was drawn with the copass point at point. The two arcs intersect at point T. b. Write two congruence stateents invoving segents in the construction. T ; T Houghton Miffin Harcourt ubishing opany c. Write a proof that the construction is true. That is, given the construction, prove T. (Hint: raw segents to create two congruent trianges.) raw. by the Refexive roperty of ongruence. T by SSS, and therefore T by T. So by the onverse of the ternate Interior nges Theore, T. 20. Use the segents shown. onstruct and abe a segent, XY _, whose ength is the average of the engths of _ and _. Justify the ethod you used. X Y Z The ength of XZ is +. The perpendicuar bisector of XZ passes through its idpoint, Y. The ength of + XY is, which is the 2 average of the given engths. Modue Lesson Lesson 6.1

10 Lesson erforance Task pastic od for copying a 30 ange is shown here. a. If you drew a triange using the od, how woud you know that your triange and the od were congruent? b. Expain how you know that any ange you woud draw using the ower right corner of the od woud easure 30. INTEGRTE MTHEMTIL RTIES Focus on Reasoning M.2 n art suppy anufacturer produces pastic trianges ike the one in the Lesson erforance Task. The ost popuar such triange has sides easuring 9.3 c, 16.1 c, and 18.6 c. with aowabe toerances of ±1%. What is the difference between the axiu and iniu aowabe perieters of the triange? axiu perieter: c; iniu perieter: c; difference: 0.88 c c. Expain the eaning of toerance in the context of drawing an ange using the od. a. The SSS ongruence Theore b. T c. ecause your penci woud be outside the triange, your triange woud be sighty arger than the triange. The toerance woud be the difference in the sizes of the two trianges. Houghton Miffin Harcourt ubishing opany Iage redits: ihaec/ Shutterstock Modue Lesson 1 EXTENSION TIVITY Have students research toerances typicay found in the anufacture of faiiar products such as batteries or eyegasses. ong topics they ay choose to investigate are effects on the product when toerances are not et; reasons a copany ay choose to epoy toerances that are greater than the ost precise; and changes in toerances ade as a product becoes ore and ore technicay precise over the years. (n Internet search for toerance chart wi get students off to a good start.) Scoring Rubric 2 points: Student correcty soves the probe and expains his/her reasoning. 1 point: Student shows good understanding of the probe but does not fuy sove or expain his/her reasoning. 0 points: Student does not deonstrate understanding of the probe. Justifying onstructions 282