Lines, Angles, and Triangles PDF
Preview Lines, Angles, and Triangles
DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-A;CA-A UNIT 8 Lines, Angles, and Triangles CONTENTS COMMON MODULE 19 Lines and Angles CORE G-CO.C.9 Lesson 19.1 Angles Formed by Intersecting Lines . . . . . . . . . . . . . . . . . . . 933 G-CO.C.9 Lesson 19.2 Transversals and Parallel Lines . . . . . . . . . . . . . . . . . . . . . . . 945 G-CO.C.9 Lesson 19.3 Proving Lines are Parallel . . . . . . . . . . . . . . . . . . . . . . . . . . 955 G-CO.C.9 Lesson 19.4 Perpendicular Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 965 G-GPE.B.5 Lesson 19.5 Equations of Parallel and Perpendicular Lines . . . . . . . . . . . . . 975 COMMON MODULE 20 Triangle Congruence Criteria CORE G-CO.B.7 Lesson 20.1 Exploring What Makes Triangles Congruent . . . . . . . . . . . . . . 989 G-CO.B.8 Lesson 20.2 ASA Triangle Congruence . . . . . . . . . . . . . . . . . . . . . . . . . 1001 G-CO.B.8 Lesson 20.3 SAS Triangle Congruence . . . . . . . . . . . . . . . . . . . . . . . . . 1015 G-CO.B.8 Lesson 20.4 SSS Triangle Congruence. . . . . . . . . . . . . . . . . . . . . . . . . . 1025 COCMOMREON MODULE 21 Applications of Triangle Congruence G-CO.D.12 Lesson 21.1 Justifying Constructions . . . . . . . . . . . . . . . . . . . . . . . . . . 1043 G-SRT.B.5 Lesson 21.2 AAS Triangle Congruence . . . . . . . . . . . . . . . . . . . . . . . . . 1053 G-SRT.B.5 Lesson 21.3 HL Triangle Congruence . . . . . . . . . . . . . . . . . . . . . . . . . . 1065 COCMOMREON MODULE 22 Properties of Triangles G-CO.C.10 Lesson 22.1 Interior and Exterior Angles. . . . . . . . . . . . . . . . . . . . . . . . 1083 G-CO.C.10 Lesson 22.2 Isosceles and Equilateral Triangles . . . . . . . . . . . . . . . . . . . 1097 G-SRT.B.5 Lesson 22.3 Triangle Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1111 COCMOMREON MODULE 23 Special Segments in Triangles G-C.A.3 Lesson 23.1 Perpendicular Bisectors of Triangles . . . . . . . . . . . . . . . . . . 1129 G-C.A.3 Lesson 23.2 Angle Bisectors of Triangles. . . . . . . . . . . . . . . . . . . . . . . . 1141 G-CO.C.10 Lesson 23.3 Medians and Altitudes of Triangles . . . . . . . . . . . . . . . . . . . 1151 G-CO.C.10 Lesson 23.4 Midsegments of Triangles . . . . . . . . . . . . . . . . . . . . . . . . . 1165 929A Unit 8 DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-A;CA-A UNIT 8 Unit Pacing Guide 45-Minute Classes Module 19 DAY 1 DAY 2 DAY 3 DAY 4 DAY 5 Lesson 19.1 Lesson 19.2 Lesson 19.3 Lesson 19.4 Lesson 19.5 DAY 6 Module Review and Assessment Readiness Module 20 DAY 1 DAY 2 DAY 3 DAY 4 DAY 5 Lesson 20.1 Lesson 20.1 Lesson 20.2 Lesson 20.3 Module Review and Assessment Readiness Module 21 DAY 1 DAY 2 DAY 3 DAY 4 Lesson 21.1 Lesson 21.2 Lesson 21.3 Module Review and Assessment Readiness Module 22 DAY 1 DAY 2 DAY 3 DAY 4 Lesson 22.1 Lesson 22.2 Lesson 22.3 Module Review and Assessment Readiness Module 23 DAY 1 DAY 2 DAY 3 DAY 4 DAY 5 Lesson 23.1 Lesson 23.2 Lesson 23.3 Lesson 23.4 Module Review and Assessment Readiness DAY 6 Unit Review and Assessment Readiness 90-Minute Classes Module 19 DAY 1 DAY 2 DAY 3 Lesson 19.1 Lesson 19.3 Lesson 19.5 Lesson 19.2 Lesson 19.4 Module Review and Assessment Readiness Module 20 DAY 1 DAY 2 DAY 3 Lesson 20.1 Lesson 20.2 Lesson 20.4 Lesson 20.3 Module Review and Assessment Readiness Module 21 Module 22 DAY 1 DAY 2 DAY 1 DAY 2 Lesson 21.1 Lesson 21.3 Lesson 22.1 Lesson 22.3 Lesson 21.2 Module Review and Lesson 22.2 Module Review and Assessment Readiness Assessment Readiness Module 23 DAY 1 DAY 2 DAY 3 Lesson 23.1 Lesson 23.3 Module and Unit Review Lesson 23.2 Lesson 23.4 and Assessment Readiness Unit 8 929B DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-A;CA-A Program Resources PLAN ENGAGE AND EXPLORE HMH Teacher App Real-World Videos Engage Access a full suite of teacher resources online and students with interesting and offline on a variety of devices. Plan present, and relevant applications of the manage classes, assignments, and activities. mathematical content of each module. Explore Activities Students interactively explore new concepts ePlanner Easily plan your classes, create using a variety of tools and approaches. and view assignments, and access all program resources with your online, customizable planning tool. Professional Development Videos Authors Juli Dixon and Matt Larson model successful teaching prac- tices and strategies in actual classroom settings. QR Codes Scan with your smart phone to jump directly from your print book to online videos and other resources. DO NOT EDIT--Changes must be made through "File info" CorrectionKey=NL-A;CA-A Teacher’s Edition Support students with point-of-use Questioning Name Class Date Strategies, teaching tips, resources for differenti- 22 . 2 Isosceles and Equilateral ated instruction, additional activities, and more. Triangles DCoOr rNeOctTio EnDKIeTy--=CNhLa-nAg;CesA m-Aust be made through "File info" DCDCoOoOr rrN erNeOcOctTtiToi Eo nEDnKDKIeTIeTy--y-=C-=CNhNhLaLa-nA-nAg;gCe;CesAsA m- Am-Auusstt b bee m maaddee t thhrorouugghh " "FFiliele i ninfofo"" Essential Question: W hat are the special relationships among angles and sides in isosceles LESSON 22 . 2 DCDCoOoOr rrNe rNeOcOctTitoT iEo nEDnKDIKeTIey-T-y-=C-=NChNhLa-LanA-ngA;gCe;CesA sAm- Am-Auusts tb bee m maaddee t hthrorouugghh " F"Filiele in infofo" " DCDCoOoOr rrNe rNeOcOctTitoT iEo nEDnKDIKeTIey-T-y-=C-=NChNhLa-LanA-ngA;gCe;CesA sAm- Am-Auusts tb bee m maaddee t hthrorouugghh " F"Filiele in infofo" " DCDCoOoOr rrNe rNeOcOctTitoT iEo nEDnKDIKeTIey-T-y-=C-=NChNhLa-LanA-ngA;gCe;CesA sAm- Am-Auusts tb bee m maaddee t hthrorouugghh " F"Filiele in infofo" " and equilateral triangles? 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Monitor student the textbook and give students step-by-step progress through reports and alerts. instructions and explanations of key math Create and customize assignments aligned to specific concepts. lessons or Common Core standards. • Practice – With dynamic items and assignments, students get unlimited practice on key concepts supported by guided examples, step-by-step solutions, and video tutorials. • Assessments – Choose from course assignments or customize your own based on course content, Common Core standards, difficulty levels, and more. • Homework – Students can complete online homework with Interactive Teacher Edition a wide variety of problem types, including the ability to Customize and present course materials with enter expressions, equations, and graphs. Let the system collaborative activities and integrated formative automatically grade homework, so you can focus where C1 assessment. your students need Lesson 19.2 Precision and Accuracy help the most! Evaluate teacher Supp1ort Lesson XX.X ComparingL eLsisnoena 1r,9 E.2x p Porenciesinotni aanl,d a Ancdcu Qraucyadratic Models • Intervention – Let 1 2 3 4 5 6 7 8 9 10 11 - 17 Personal Math Trainer EXPLAIN Concept 1 Explain CCCCoooonnnncccceeeepppptttt 2222 the Personal Math Question 3 of 17 View Step by Step Video Tutor Textbook X2Animated Math CSIoomNPSLEtINg thE SquArEDDDD wItheeee EXPttttrEeeeeS-rrrrmmmmiiiinnnniiiinnnngggg PPPPrrrreeeecccciiiissssiiiioooonnnn Solve the quadratic equation by factoring. 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Check Turn It In Look Back Focus on Higher Order Thinking Raise the bar with homework and practice that incorporates higher-order thinking and mathematical practices in every lesson. Differentiated Instruction Resources Support all learners with Differentiated Calculate the minimum and maximum possible areas. Round your answer to the nearest square centimeters. Assessment Readiness Instruction Resources, including The width and length of a rectangle are 8 cm and 19.5 cm, respectively. Prepare students for success on high • Leveled Practice and Problem Solving Find the range of values for the actual length and width of the rectangle. stakes tests for Integrated Mathematics 1 • Reading Strategies Minimum width = 7 . 5 cm and maximum width < 8 . 5 cm with practice at every module and unit • Success for English Learners My answer • Challenge Find the range of values for the actual length and width of the rectangle. Assessment Resources Minimum length = 1 9 . 4 5 cm and maximum length < 1 9 . 5 5 cm Tailor assessments and response to intervention to meet the NLaE1mS-Se1O _ N_ ___PS_u_r_ce_c_c_e_iss__si_o _f_on_r_ _aE__nn_g_d_l_ i_Ss_h_i_g L__ne_ai_f_ri_nc_e_a_r_ns Dt aDte i_g__i_t_s__ ___________ Class __________________ NLaE6mS-S1eO N__ __RL__ei_nt_e_e_a_ac_h_r_ _F__u__n__c_t_i_o__n__s__ __________ Date __________________ Class __________________ needs of all your classes and students, including BiTfPCPCsrhae hhrrmeccoooota oooipobburssrenseeell eee cpott ihhmmrfas eeeai och mmniu4u12sn n2neoooe i.d.rrtfa3 ee ruar e gepps dm serrittseehde tcc .taeo oiisssnf ee uttahh rmm ee.g meeraaa4emss2nuu. t3i rrsiee s gsmm d moeeera nnt e4lttl.:re2 m3r. 26itnh7 ieFa ngidnnc hdbae4 y tts2 het .hehon2 eurtrn7h a ne3 s ngdoa mgf rfreie seaeaed lsot ltteogt.hf s rv.tat hRRa muelImm unn ,e eeieet4tsaah 2o ssfe .oruuflfl 2 arrr7 eebt mmho geeev ee eann ccceMtt t.sx u Eierttnaxericlpsm ilul se aul entimn,sg t ihth nA h oe =r d wael ionaff tcd e=oa r7w tefiMi.onix5din tn ttc h aihommmi fso u t×opfhm u tr1ehon a 9TTwebut1l.ehs wln 4ir e. dle eo c5 almE t _c e1f ywhg _cxAt rT u.4s ar _m t a •h× x •anh an_ a m e = peci_ gM+n te _h vTl pAhtg2e_Bri eynhltor_e,.ey a raie_ifrBmtn sxi p_ n+iyaeva, ch_hut t raa_rahC=aay eme_orcn e lv iiC _f=annhd enlT ll_iae i bno o_nh0tiCfens leth tu eel e x g ii.iemaasnsanns pt r l 2rc ceehowutx xdexx6ah2yt an yly ea ai2tlae =rroeci 2 =+nypre=n= =u nn.efs s8ola 3 d 4uilt9 _8 t 5ea m3sl Nia a n_ y ts=d xn _neioOc h nyu _6ldtdfti oaoT_ mi n= 1ao_gtr oe blr_in2e izdra_en7t so_hi rres f,_ sne ooa _ey.yatrx rr_rx xea A n m i _i snpifxnlnoa n_ uta op l or n_ aifenvanndnoo_td xaci.edes enrg pr yqt. nnethi Bioah uTootnmtsn be amthnal eil urooenerseienlen3rn e t:q gti a.rr p .un xro at laao_1o iode≥_ptt4rtii d _ c ohb1_= an ot _ool t _2ofhxsg _8fxi e_ gz a_t =neh _ lsre_iCon._r . e _ is_a _r f unctio4n. ._ 6 _x_2_ _=_ 1__2_ ______ ••••• LLUPTeeilneavvirceet e1 llPeem, eTddrei feMUnortnr o,2m idDt, aauTiaennlegscd tneQ sTo Tuiesaitrzsi zkc3e,s Rasensdo Qurucaersterly Benchmark Tests Examples 1. cWohnesnid deer?c iding which measurement is more precise, what shoFFuoolrrdmm yuuolluaa SSeenndd ttoo NNootteebbooookk Unit 8 929D _________________________________________________________________________________________ 2. An object is weighed on three different scales. The results are shown O riginailn c Sotnhcte123ean t l tCeao pbylreig.h tMW © eb1h1ya12i H2cs2.oh.4uu.5 56grs3he 2ct oom oanzo zlM zee i fn fliisnt HthaErecxo___ pum___rlto___. orA___eds___dt___i t___ipo___nrs___e ___acn___ids___ c___eh6___?a ___n g___Ee___sx___ tpo___ l___tah___ei___ no___r iy___gi___ona___ul___ cr___o an___t___enn___st___ wa___re___e t___rh.___e ___re___s___p___on___s___ib___il___ity___ o___f ___th___e___ i___ns___tr___u___ct___o___r. _______________ O orTig5rein .lan l l_ 9 ecw_oy in _t h th_=e eyn_ et2 _ tCr=h_7.o _ep 1_yr r_i ge_h_at ©_c _bh_y _He oquguhtaont i6Mo.ifn fl _i6n _r xH_e ax_+pr_c =ro_7ue _ry2ts_. A_e=d_n d1_itt_0iso_n _sa _a n_hd o c8hr7ai nzgoesn7 tto.a t h _le 2y_1 loi _rn=xi_g ei_n−=_,a3 l_1 ac _o9 _nvt _ee _nrt_ ta_iYrce_o _tauh elr Trleiusnrpneon,s ibili t_xy8 _ o.=f_ tx _h3e _= _in _s0t_r u_c_to_r._ ___ DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-A;CA-A Math Background Parallel Lines COMMON GGG---CCCOOO...CCC...999 The point at which the perpendicular segment through P CORE intersects line is sometimes called the foot of the point on ℓ LESSONS 19.1 to 19.3 the line. The Parallel Postulate was the fifth postulate proposed in Euclid’s Elements. Euclid worded the postulate as follows: If Congruent Triangles COMMON GGG---CCCOOO...BBB...777 CORE a straight line falling on two straight lines makes the interior LESSONS 20.1 to 20.3 angles on the same side less than two right angles, then the straight lines, if extended indefinitely, meet on the side on Two geometric figures are congruent if they are the same which the angles lie. size and shape; in other words, if one of the figures can be moved so that it fits perfectly on top of the other figure. Today, the postulate is usually presented in a logically equivalent form that is sometimes known as Playfair’s This is the intuitive idea behind the more rigorous Axiom: Through a point P not on line , there is exactly one line mathematical definition of congruence: two figures are ℓ parallel to congruent if one can be transformed into the other by ℓ. an isometry (that is, by a combination of translations, The Parallel Postulate has played an important role reflections, and rotations). in the history of mathematics, initially because many mathematicians believed it was actually a theorem that For polygons, the definition of congruence can be stated in could be proved from Euclid’s first four postulates. It was terms of corresponding sides and angles. In particular, two only in the nineteenth century that Eugenio Beltrami triangles are congruent if and only if the sides and angles proved that the Parallel Postulate could not be proven from can be matched up so that the corresponding sides are Euclid’s four other axioms. congruent and the corresponding angles are congruent. This definition of triangle congruence means that six Perpendicular Lines COMMON GGG---CCCOOO...CCC...999 CORE correspondences must be checked in order to conclude that LESSONS 19.4 to 19.5 two triangles are congruent (three pairs of corresponding sides and three pairs of corresponding angles). The Thus far, the concept of distance has been defined only congruence theorems provide shortcuts for proving for two points. However, it is possible to extend the notion triangles congruent. of distance to other situations. For example, the distance from a point P to a line is defined as the length of the ℓ Strictly speaking, only one of SSS, SAS, or ASA needs to be perpendicular segment from P to . ℓ taken as a postulate. In other words, any one of these may be assumed to be true, and the other two may then be As shown in the following figure, this perpendicular proved as theorems. segment is the shortest segment from the point to the line. Euclid actually proved all three results (SSS, SAS, and ASA) P as theorems. However, he did so through the use of a “superposition” postulate that allowed one triangle to be placed on top of another. ℓ Modern mathematicians consider this type of motion to The foot of point p on line ℓ . be invalid within the logical system of classical Euclidean geometry, and for this reason one of the three statements is generally taken as a postulate. 929E Unit 8 DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-A;CA-A PROFESSIONAL DEVELOPMENT Properties of Triangles COMMON GGG---CCCOOO...CCC...111000 In this sense, the centroid can be considered the “average” CORE of all the points in the triangle. The triangle will also LESSONS 22.1 to 22.3 balance along any line that passes through the centroid. In The Triangle Inequality is the mathematical statement of a particular, this means that a median of a triangle divides the well-known fact: the shortest path between two points is a triangle into two smaller triangles with equal areas. straight line. In the case of a triangle with vertices A, B, and For example, in the figure below, C, the straight path from A to B is shorter than the path that area (△ABX) = area (△ACX) . includes a detour to point C. In other words, AB AC BC. < + A One can use the Triangle Inequality to find other useful results. For example, the length of any side of a triangle is less than half the perimeter of the triangle. Another Q result is that the sum of the lengths of the diagonals of any quadrilateral is less than the perimeter of the quadrilateral. Specifically, in quadrilateral ABCD, four applications of the B X C Triangle Inequality yield the four inequalities shown below. This is easy to see because the two smaller triangles have equal bases and heights. AC AB BC A B < + Midsegments of Triangles COMMON GGG---CCCOOO...CCC...111000 AC AD CD CORE < + LESSON 23.4 C BD AD AB < + A midsegment of a triangle is a segment that joins the D BD CD BC < + midpoints of two sides of the triangle. Together, the three midsegments of a triangle form the midsegment triangle, which is XYZ in the figure. △ Adding the four inequalities above gives A 2 AC BD 2AB 2BC 2CD 2AD , and dividing both ( + )< + + + sides of this inequality by 2 proves the result. Z Y Special Segments in Triangles COMMON GGG---CCCOOO...CCC...111000 CORE B X C LESSONS 23.1 to 23.3 By the Triangle Midsegment Theorem, The medians of a triangle are concurrent at a point called the centroid of the triangle. The centroid has important XY = _ 12 AB, ZY = _ 12 BC, and XZ = _ 12 AC. It follows by SSS Congruence that the four small triangles physical properties. For a triangle of uniform thickness and formed by the midsegments are congruent. density, the centroid is the point at which the triangle will balance. Since the four triangles together form ABC, each small △ triangle has one-fourth the area of ABC. △ Unit 8 929F DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-A;CA-A DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-A;CA-A 8 UNIT Lines, Angles, and UNIT 8 MODULE 19 Lines, Angles, Lines and Angles Triangles 20 MODULE Triangle Congruence and Triangles Criteria 21 MATH IN CAREERS MODULE Unit Activity Preview Applications of Triangle Congruence After completing this unit, students will complete a 22 MODULE Math in Careers task by applying triangle congruence Properties of Triangles theorems to the real world. Critical skills include 23 MODULE modeling real-world situations. Special Segments in Triangles For more information about careers in mathematics as well as various mathematics appreciation topics, visit The American Mathematical Society at http://www.ams.org. Ocean/ © mage Credits: mpany • I MATH IN CAREERS Harcourt Publishing Co Afaduonesrretdc a d mhilelii vsaateitnegh. dncIen timsn tagraAdo tsnidnpc igsaat irctaocoernhs tc iiwtstroetehic acaett erskiek ses iperplelnaessc o,pe eapyosrle nect hhsfwoiiabtrotel reack rts e © Houghton Mifflin Corbis bIamfro caythtohh•iu t feAe’urmcenltg ac,i etntyibitcooeraunarle assslhut aeobndujed licd nat sesa:ts utchdaeryet itechra ealslsye apnle asing. • Geometry • Trigonometry Research other careers that require the use of spatial analysis to understand real- world scenarios. See the related Career Activity at the end of this unit. Unit 8 929 TRACKING YOUR LEARNING PROGRESSION IN1_MNLESE389762_U8UO 929 4/19/14 10:55 PM Before In this Unit After Students understand: Students will learn about: Students will study: • using the distance formula on a coordinate plane • parallel lines, transversals, and angle relationships • properties of intersecting lines, parallel lines, and • constructing an angle bisector • perpendicular lines and bisectors perpendicular lines in figures • postulates about segments, angle, lines, and planes • slopes and equations of parallel and • properties of quadrilateral figures: parallelograms, • rigid and non-rigid motions perpendicular lines rectangles, rhombuses, squares, kites, and trapezoids • congruence of corresponding parts • congruence of triangles • perimeter and area on the coordinate plane • geometric constructions • special triangles and triangle inequalities • special segments of triangles 929 Unit 8 DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-A;CA-A DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-A;CA-A Reading Start -Up Reading Start Up Vocabulary Review Words ✔ adjacent angles (ángulos Have students complete the activities on this page by Visualize Vocabulary adyacentes) working alone or with others. Use the ✔ words to complete the case diagram. Write the review words in ✔ parallel lines (líneas the bubbles and draw a picture to illustrate each case. paralelas) ✔ congruence (congruencia) Angle ✔ vertical angles (ángulos VISUALIZE VOCABULARY Relationships verticales) ✔ complementary angles The case diagram graphic helps students review (ángulos complementarios) vocabulary associated with angles. If time allows, vertical supplementary ✔ supplementary angles angles angles (ángulos suplementarios) review relationships among angles created by parallel complementary ✔ transversal (transversal) angles lines and a transversal. Preview Words indirect proof (demostración indirecta) hypotenuse (hipotenusa) UNDERSTAND VOCABULARY legs (catetos) interior angle (ángulo interior) Use the following explanations to help students learn exterior angle (ángulo exterior) the preview words. isosceles triangle (triángulo isósceles) equilateral triangle (triángulo A triangle with three sides that are the same length equilátero) Understand Vocabulary circumscribe (circunscrito) is equilateral. A triangle with two sides that are inscribed (apuntado) the same length is isosceles. A triangle with a right Complete the sentences using the preview words. equilateral triangle angle is a right triangle. The sides of a right 1. A(n) has three sides with the same length. inscribed triangle that form the right angle are the legs. The 2. A circle is in a polygon if each side of the polygon is tangent to the circle. side opposite the right angle is the hypotenuse. An 3. The hy po te n us e of a right triangle is the longest side of the triangle. © Houghton M ionf ttehrei otrri aannggllee wofi tah t ar icaonmglme ios nfo vremrteedx .b Ay ntw eox tseirdieosr ifflin H angle is formed by one side of the triangle and an Active Reading arcourt Publishing Com ACexTtIeVnsEio nR oEf AanD aIdNjaGcent side. Ktoe hye-lTpe yromu Foorgladn iWze hviolec arbeuadlairnyg weaocrhd sm. Wodruitlee ,v corceaabteu laa rkye yte-rtemrms o fno ld pany Students can use these reading and note-taking one side and definitions on the other side. Place a special emphasis strategies to help them organize and understand the on learning and speaking the English word while discussing the unit. new concepts and vocabulary. Encourage students to ask questions to create definitions that are clear, correct, and helpful. Remind them to include Unit 8 930 diagrams to support their definitions. It may be beneficial to have students share information to help clarify definitions of any terms that seem IN1_MNLESE389762_U8UO 930 4/19/14 11:32 AM confusing. ADDITIONAL RESOURCES Differentiated Instruction • Reading Strategies Unit 8 930 DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-A;CA-A DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-A;CA-A 19 MODULE MODULE Lines and Angles 19 Lines and Angles ESSENTIAL QUESTION: Answer: The characteristics of parallel and Essential Question: How can you use parallel and LESSON 19.1 perpendicular lines can help you to analyze perpendicular lines to solve real-world problems? Angles Formed by Intersecting Lines real-world objects such as street intersections. LESSON 19.2 Transversals and Parallel Lines This version is for PROFESSIONAL DEVELOPMENT Algebra 1 and VGIeoDmEetrOy only LESSON 19.3 Proving Lines Are Parallel Professional Development Video LESSON 19.4 Author Juli Dixon models successful Perpendicular Lines teaching practices in an actual high-school classroom. LESSON 19.5 Equations of Parallel and Perpendicular Lines Professional Demveyl.ohprwm.ceonmt Alexander © mage Credits: © Houghton Mifflin Harcourt Publishing Company • IDemianchuk/Reuters/Corbis MMODyUsLEt PeERrFyOR MSApNCoE TtA SBK uPRiElVdIEWingRCacilalnhEundeAs cibpLokee n Wo ruspu Osietne Rnh daLdo tmDiwoc uy VcpslraIrteDoera rplEtiyeneO serr ptesi oeaasltn- wboduf o apirlndladgrin alegllse. l In this module, you will use properties of parallel lines and angles to analyze the strange happenings in a mystery spot building. With a little bit of geometry, you’ll be able to figure out whether mystery spot buildings are “on the up-and-up!” Module 19 931 DIGITAL TEACHER EDITION PERSONAL MATH TRAINER IN1_MNLESE389762_U8M19MO 931 4/19/14 11:48 AM Assessment and Intervention Access a full suite of teaching resources when and where you need them: Assign automatically graded homework, quizzes, • Access content online or offline tests, and intervention activities. Prepare your students with updated, Common Core-aligned • Customize lessons to share with your class practice tests. • Communicate with your students in real-time • View student grades and data instantly to target your instruction where it is needed most 931 Module 19 DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-A;CA-A DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-A;CA-A Are YOU Ready? Are You Ready? Complete these exercises to review skills you will need for this chapter. ASSESS READINESS Angle Relationships Example 1 B The measure of ∠AFB is 70° and the measure •• OHinnltins ea nHdo mHeelwpork Use the assessment on this page to determine if C of ∠AFE is 40°. Find the measure of angle ∠BFE. • Extra Practice students need strategic or intensive intervention for A m∠BFE = m∠AFB + m∠AFE Angle Addition Postulate the module’s prerequisite skills. F m∠BFE = 70° + 40° Substitute. m∠BFE = 110° Solve for m∠BFE. E D ASSESSMENT AND INTERVENTION Find the measure of the angle in the image from the example. 1. The measure of ∠BFE is 110°. Find m∠CFD. m∠EFD = 7 0° 2. The measure of ∠BFE is 110°. Find m∠EFD and m∠BFC. m∠BFC = 7 0° Parallel Lines Cut by a Transversal t Example 2 The measure of ∠7 is 110°. Find m∠3. 1 2 p 321 TIER 1, TIER 2, TIER 3 SKILLS Assume p∥q. 4 3 Personal Math Trainer will automatically create a m∠3 = m∠7 Corresponding Angles Theorem 5 6 standards-based, personalized intervention m∠3 = 110° Substitute. 8 7 q assignment for your students, targeting each student’s individual needs! Find the measure of the angle in the image from the example. Assume p∥q. 3. The measure of ∠3 is 110°. Find m∠1. m∠1 = 1 1 0° © 4Wa E . nxardmiTtph iHleen m3og era sEiuzrqeo uoF (n (fyi ayn ∠ty– –dt a3 –y i 6t 1lioh6ys)) eL1====n 1lii 0snnm222°e( xx .( eox xpF– –s afi –n6 r 3 daPx)l 1l m)ae l∠ rtUSSSo6a uio s.my lbel vl= pspeetl oi ift2lfoiuy,xnr .t m tP +ey- ∠s.fe lo76o rr tp =hmpea ,e tf x op n 1ra ,m sd y s .e1 i .s cP tauh rralo 7laul0e gr°lh ,l i tnVh eees phroa tivni etc (tah3,le ,6 s )a. me slope, so m = 2. Houghton Mifflin Harcourt Publishing Company SrRAeees••Deso ptuTTDohriineceeI serrTtea s22 Ibta OolSSve kkaI NbniiillletllaAe lbPPorlLorvweese -ftnRfoT-otTreEri eos taStshns tfi O sRsuf o flmeoUlrs r looei ResuadtacCru cohclheEfe M s .isS nakotliesdlolruv lieennctliuodne s: Find the equation of the line described. 5. Perpendicular to y = 3x + 5; passing through the point ( –6, –4) y = - _ 13 x - 6 6. Parallel to the x-axis; passing through the point ( 4, 1) y= 1 Module 19 932 IN1_MNLESE389762_U8M19MO 932 Response to Intervention Differentiated 4/19/14 11:47 AM Instruction Tier 1 Tier 2 Tier 3 Lesson Intervention Strategic Intervention Intensive Intervention Worksheets Skills Intervention Worksheets available Worksheets online Reteach 19.1 34 Angle Relationships Building Block Skills 7, Challenge Reteach 19.2 40 Parallel Lines Cut by a 15, 16, 22, 23, 42, 46, worksheets Reteach 19.3 Transversal 53, 56, 66, 71, 87, 95, Extend the Math Reteach 19.4 42 Properties of 98, 102, 103 Lesson Activities Reflections in TE Reteach 19.5 47 Writing Equations... Module 19 932
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