Overview Describe Angle Relationships
LESSON 6Overview Describe Angle RelationshipsSTANDARDS FOR MATHEMATICALPRACTICE (SMP)ObjectivesVocabularySMP 1, 2, 3, 4, 5, and 6 are integrated into theTry-Discuss-Connect routine.*Content Objectives Math VocabularyThis lesson provides additional support for:3 Construct viable arguments and critiquethe reasoning of others.* See page 1o to learn how every lesson includesthese SMP. Identify corresponding angles, alternateinterior angles, alternate exterior angles,same-side interior angles, and same-sideexterior angles when given a pair of linesthat is cut by a transversal.Informally establish and understand theangle relationships that exist whenparallel lines are cut by a transversal.Use angle relationships to find unknownmeasures of angles formed by parallellines cut by a transversal.Use angle relationships to determinewhether two lines cut by a transversalare parallel.Language Objectives Confirm understanding of lesson vocabulary by identifying examplesof angle relationships and namingappropriate angles with geometricnotation.Explain whether angles are congruentby analyzing geometric figures andusing definitions during discussion andin writing.Justify angle measurements by namingthe appropriate angle relationship.Understand and use lesson vocabularyand reasoning to explain why two linescut by a transversal are parallel.Listen to and compare ideas with a partnerand provide reasons for any disagreement.Prior Knowledge Understand that vertical anglesare congruent.Know that the measure of a straight angleis 180 .Know that the sum of the measures ofsupplementary angles is 180 .Recognize parallel lines.alternate exterior angles when two linesare cut by a transversal, a pair of angles onopposite sides of the transversal andoutside the two lines.alternate interior angles when two linesare cut by a transversal, a pair of angles onopposite sides of the transversal andbetween the two lines.corresponding angles angles in the samerelative position when two lines are cut by atransversal.linear pair two angles that are adjacentand supplementary.same-side exterior angles when two linesare cut by a transversal, a pair of angles onthe same side of the transversal and outsidethe two lines.same-side interior angles when two linesare cut by a transversal, a pair of angles onthe same side of the transversal andbetween the two lines.transversal a line that cuts two or morelines. The lines cut by the transversal may ormay not be parallel.Review the following key terms.adjacent angles two non-overlappingangles that share a vertex and a side.supplementary angles two angles whosemeasures sum to 180 .vertical angles opposite angles formedwhen two lines intersect. Vertical angles arecongruent.Academic Vocabularyintersect to meet or cross. When two linesintersect, they cross at a common point.Learning ProgressionEarlier in Grade 8, students learnedthat both parallel lines and anglemeasures are preserved by rigidtransformations. Students also usedthe relationships between verticalangles and linear pairs of angles to findangle measures.115aLESSON 6 Describe Angle RelationshipsIn this lesson, students establish factsabout the relationships among themeasures of angles formed by twoparallel lines cut by a transversal.Students also learn how to find newangle relationships using the anglerelationships they already know.Later in Grade 8, students will use therelationships discovered in this lessonto discover and show relationshipsamong the interior and exterior anglesof a triangle. Curriculum Associates, LLCCopying is not permitted.
LESSON 6OverviewPacing GuideItems marked withSESSION 1 are available on the Teacher Toolbox.MATERIALSDIFFERENTIATIONExplore Angle Relationships (35–50 min)Start (5 min) Try It (5–10 min )Discuss It (10–15 min)Connect It (10–15 min)Close: Exit Ticket (5 min)Presentation SlidesPREPARE Interactive TutorialRETEACH or REINFORCE Hands-On ActivityMaterials For each student: protractorAdditional Practice (pages 119–120)SESSION 2 Develop Describing Congruent Angle Relationships (45–60 min)Start (5 min) Try It (10–15 min )Discuss It (10–15 min)Connect It (15–20 min)Close: Exit Ticket (5 min)Math Toolkit graph paper, tracingpaper, transparenciesPresentation Slides REINFORCE Fluency & Skills PracticeEXTEND Deepen UnderstandingAdditional Practice (pages 125– 126)SESSION 3RETEACH or REINFORCE Hands-On ActivityMaterials For each student: 4 markers,index card, ruler, scissorsDevelop Describing Supplementary Angle Relationships (45–60 min)Start (5 min) Try It (10–15 min )Discuss It (10–15 min)Connect It (15–20 min)Close: Exit Ticket (5 min)Presentation SlidesRETEACH or REINFORCE Hands-On ActivityMaterials For each student: protractor, rulerREINFORCE Fluency & Skills PracticeEXTEND Deepen UnderstandingAdditional Practice (pages 131– 132)SESSION 4 Refine Describing Angle Relationships (45–60 min)Start (5 min) Monitor & Guide (15–20 min )Group & Differentiate (20–30 min)Close: Exit Ticket (5 min)Math Toolkit Have items fromprevious sessions available forstudents.Presentation SlidesRETEACH Hands-On ActivityMaterials For each student: 2 colored pencils(1 blue, 1 red), centimeter ruler, scissors,tracing paperREINFORCE Problems 4–9EXTEND ChallengePERSONALIZELesson 6 Quiz orDigital Comprehension CheckRETEACH Tools for InstructionREINFORCE Math Center ActivityEXTEND Enrichment Activity Curriculum Associates, LLCCopying is not permitted.LESSON 6 Describe Angle Relationships115b
LESSON 6Overview Describe Angle RelationshipsConnect to Culture Use these activities to connect with and leverage the diverse backgroundsand experiences of all students. Engage students in sharing what theyknow about contexts before you add the information given here.SESSION 1Try ItAsk students to raise their hands if they have ever checked the status of aflight using a computer or phone app. Have some students share their experiences.It is important for anyone in the aviation industry to be able to track flights. Everyairplane uses a computerized system to transmit information such as altitude,direction, speed, and precise latitude and longitude positions during any portion ofits flight. There are several flight-tracking software systems available to those in theindustry as well as the general public. Most flight trackers get their flight data fromthe United States Federal Aviation Administration, AirNav Systems, and GlobalDistribution Systems, as well as from direct airport data feeds. While the generalpublic may use these tracking systems to see when a flight of a relative or friendmay arrive, those in the industry use these systems to ensure the safety ofpassengers and accountability of those operating airplanes.Flight Information1:59 PMNWESSESSION 2Try ItAsk if any students have ever been to a Native American wedding. The vaseshown is a wedding vase from the Taos Pueblo, a federally recognized tribal nationof Native Americans in Taos, New Mexico. Native American wedding vases have twospouts that symbolize the two people getting married. The handle in the middlesymbolizes the union of the couple. The open space between the handle and spoutsrepresents the circle of life. As part of the Native American wedding ceremony, thecouple drinks from their wedding vase at the same time. Parents of the groom in theceremony typically provide the wedding vase, either by buying one or making onethemselves. Married couples normally keep their wedding vase on display after thewedding. Ask students if they have been to a wedding in which a similar symbolicunion of the couple occurred. Have them share their experiences with the class.SESSION 3Try ItAsk if any students have ladder shelves in their homes. Ladder shelves canbe used for many purposes. Some are used to display plants, either indoors oroutdoors. Other ladder shelves are used to display photographs, such as ones froma family vacation. Some ladder shelves are used as exercise apparatuses for pets.Cats can climb up and down the shelves and rest as needed. Have students draw adiagram of a ladder shelf they would design to have in their home. Select studentsto share their drawings.CULTURAL CONNECTIONAlternate Notation You can use angle notation to write “angle ABC”as /ABC. In many Latin American countries, different notation is usedwhere the angle symbol is placed above the letters instead of to theleft. Encourage students who have experience with this angle notationto share what they know with the class.115cLESSON 6 Describe Angle RelationshipsC/ABR—O —ABC Curriculum Associates, LLCCopying is not permitted.
LESSON 6OverviewConnect to Family and Community After the Explore session, have students use the Family Letter to let theirfamilies know what they are learning and to encourage family involvement.LESSON 6 DESCRIBE ANGLE RELATIONSHIPSActivity Thinking AboutAngle RelationshipsLESSON6This week your student is learning about angle relationships. Angles are formedwhen two lines are intersected, or cut, by a third line. The third line is called atransversal. When the two lines are parallel, some of the angles formed by thetransversal are congruent.klkl klAB is parallel to ·CD . ·EF is the transversal. Three types ofIn the figure below, ·congruent angles are formed when parallel lines are cut by a transversal: /4 and /5 are alternate interior angles.Alternate interior angles are on opposite sidesof the transversal and between the two linescut by the transversal.AC1E23 4 /1 and /8 are alternate exterior angles.Alternate exterior angles are on the oppositesides of the transversal and outside the twolines cut by the transversal.6587FDB /2 and /6 are corresponding angles. Corresponding angles are in the samerelative position when two lines are cut by a transversal. Do this activity together to investigateangle relationships in real life.There are many places in the world aroundyou where angles and their relationships areimportant. One example is a truss bridge.Part of the structure of a truss bridge is shown.These bridges are built using a designinvolving parallel lines cut by transversals.The angles formed by this structure meetthe required safety standards for strengthand stability!Describe Angle RelationshipsDear Family,What angle relationships do you see in the picture of the truss bridge?What are other real-world examples of angle relationships?Your student will learn to use angle relationships to identify angle measurements.klklIn the figure below, ·UV is parallel to ·WX . Can you name a pair of angles that havethe same measure? ONE WAY to use angle relationships is to identifySalternate interior angles.V/3 and /6 are alternate interior angles.m/3 and m/6 are equal. ANOTHER WAY to use angle relationships is to21 43UX65 87identify alternate exterior angles./1 and /8 are alternate exterior angles.m/1 and m/8 are equal.TWUse the next page to start a conversationabout angle relationships.LESSON 6 Describe Angle Relationships Curriculum Associates, LLC Copying is not permitted.115116LESSON 6 Describe Angle Relationships Curriculum Associates, LLC Copying is not permitted.Connect to Language For English language learners, use the Differentiation chart to scaffold thelanguage in each session. Use the Academic Vocabulary routine for academicterms before Session 1.DIFFERENTIATION ENGLISH LANGUAGE LEARNERSUse with Session 1Connect ItLevels 1–3: Listening/WritingLevels 2–4: Listening/WritingLevels 3–5: Listening/WritingHelp students prepare to respond to ConnectIt problem 2. Read the problem aloud asstudents follow along. Use a Co-ConstructedWord Bank to help students understandthe language in this problem. Have studentscircle words that they are unsure of. Pointout positional words such as opposite, inside,between, outside, and relative to.Write these words on the board and havestudents listen as you represent each wordwith a diagram using simple symbols such asstars and hearts to model which symbols arebetween two lines, outside of the lines, and inthe same position relative to the lines. Havestudents add these words and the diagramsto their word banks, along with any otherwords they circled.Help students prepare to solve Connect Itproblem 2. Read the problem with students.Use a Co-Constructed Word Bank tohelp students write about pairs of angles.Have students circle words that name anddescribe pairs of angles. Use the words tostart a word bank on the board.Point out positional words in the problemsuch as opposite, inside, between, outside, andrelative to that describe the angles. Modelhow to draw a diagram of the pair of anglesas they listen to the positional terms. Guidestudents to represent each pair of angles intheir word bank with diagrams.Prepare students to write responses toConnect It problem 2. Have students readthe problem. Use a Co-Constructed WordBank to help students represent pairs ofangles. Have students circle words that nameand describe pairs of angles. Then havethem use the words to create a word bank.Have students add diagrams to representeach word. Ask students to turn and talk to apartner to review the word banks and makesure they used relevant words and sketchedaccurate diagrams. Have them refer to theword banks as they write about pairs of anglesin this lesson. Curriculum Associates, LLCCopying is not permitted.LESSON 6 Describe Angle Relationships115–116
LESSON 6 SESSION 1Explore Angle RelationshipsPurpose Explore the idea that angle relationships can be used tofind unknown angle measures in a given figure.Understand the different types of related anglepairs that are formed when a pair of lines is cut bya transversal.STARTCONNECT TO PRIOR KNOWLEDGEStartAlways, Sometimes, NeverBA 45 -angle and a right angle arecongruent.Supplementary angles are congruent.CVertical angles are congruent.DAdjacent angles are congruent.ALESSON 6 SESSION 1Flight Information1:59 PMExplore Angle RelationshipsNPreviously, you learned about pairs of angles formed whentwo lines intersect. In this lesson, you will learn about pairsof angles formed when one line intersects two other lines.WES Use what you know to try to solve the problem below.Zahara says she can use angle relationships to find all the anglemeasures in the figure. What is m/BCF?AC (x 1 15) BD2x EF x 55 HG Curriculum Associates, LLC Copying is permitted.SolutionsTRYITA is never true.Possible work:B is sometimes true.SAMPLE AC is always true./EFH and /CFG are vertical angles, so x 5 55. This means 2x 5 110./DCF and /BCF are supplementary angles.D is sometimes true.2x 1 m/BCF 5 180 DISCUSS IT110 1 m/BCF 5 180 WHY? Support students’ understanding ofangle relationships.m/BCF 5 70 Ask: How did youdecide which anglemeasure to find first?SAMPLE BVertical angles are congruent. So, x 5 55 and m/BCF 5 (x 1 15) .TRY ITSMP 1, 2, 4, 5, 6Make Sense of the Problem(x 1 15) 5 (55 1 15) 5 70 m/BCF 5 70 Share: The first anglemeasure I foundwas . . .Learning TargetSMP 1, SMP 2, SMP 3, SMP 4, SMP 5, SMP 6Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the anglescreated when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles.See Connect to Culture to support studentengagement. Before students work on Try It, useCo-Craft Questions to help them make sense of theproblem, showing only the diagram. Students willlikely suggest finding the value of x as a possiblequestion but may also mention determining themeasure of other specific angles in the figure.Common Misconception Listen for students who think the 55 angle and /ACDhave the same measure. As students share their strategies, have them trace the55 angle and place their tracing over /ACD so they can see those angles arenot congruent.DISCUSS ITSelect and Sequence Student StrategiesSMP 2, 3, 6Support Partner DiscussionAfter students work on Try It, have themrespond to Discuss It with a partner. Listen forunderstanding that: /CFG is a vertical angle with the 55 angle, so thevalue of x can be found. /ACD and /DCF are supplementary angles, soan equation can show the sum of those anglemeasures is 180 , and the value of x can be found.117LESSON 6 Describe Angle RelationshipsLESSON 6 Describe Angle Relationships Curriculum Associates, LLC Copying is not permitted.117117Select 2–3 samples that represent the range of student thinking in your classroom.Here is one possible order for class discussion: vertical angles used to find x, expression x 1 15 used to find m/ACD, and verticalangles used to find m/BCF (misconception) incorrect congruent angles identified, leading to anincorrect solution vertical angles used to find x, expression 2x used to find m/DCF, and supplementary angles used to find m/BCF Curriculum Associates, LLCCopying is not permitted.
LESSON 6 SESSION 1ExploreFacilitate Whole Class DiscussionLESSON 6 SESSION 1Call on students to share selected strategies. Remindstudents that one way to agree and build on ideas isto give reasons that explain why the strategy makessense. Invite students to reword informal languagewith mathematical vocabulary.CONNECT ITLook Back What is m/BCF? What types of angle relationships did you use to1find m/BCF?70 ; Possible answers: supplementary angles, vertical anglesGuide students to Compare and Connect therepresentations. Call on several students to rephraseimportant ideas so that everyone hears them morethan once and in more than one way.Look Ahead The figure in the Try It problem shows pairs of angles you know,such as adjacent angles, supplementary angles, and vertical angles. The figure alsoshows pairs of angles that are new to you.2a. You know that supplementary angles are two angles whosemeasures have a sum of 180 . A linear pair is a pair ofsupplementary angles that are adjacent. What two angles formthe linear pair shown? What is the value of x?ASK What angle relationships do all of thesestrategies use?LISTEN FOR The strategies use the fact thatvertical angles have the same measure and thatthe sum of the measures of supplementary anglesis 180 .CONNECT IT1Dbca132457 68d. /3 and /6 are alternate exterior angles. These angles are onopposite sides of the transversal, but are on the outside of the othertwo lines. What is the other pair of alternate exterior angles?/1 and /8e. /1 and /5 are corresponding angles. These angles are in the same positionrelative to the lines and the transversal. /2 and /6 are also correspondingangles. What are the other two pairs of corresponding angles?/3 and /7, /4 and /8DIFFERENTIATION RETEACH or REINFORCEHands-On ActivityReflect Is it possible for a pair of angles to be both corresponding angles andalternate interior angles? Explain.3 Verify angle measures with aprotractor.Copying is not permitted.Cc. In the figure, /2 and /7 are alternate interior angles. These anglesare on opposite sides of the transversal, and they are inside, orbetween, the other two lines. What is the other pair of alternateinterior angles? /4 and /5relationship between vertical angles can beused to find x, and then one of the expressions,x 1 15 or 2x, and the relationship betweenvertical or supplementary angles can be usedto find m/BCF. Curriculum Associates, LLCBb. A transversal is a line that intersects or cuts two or more lines. Whichline is the transversal in the figure at the right? line aLook Back Look for understanding that theMaterials For each student: protractor Instruct students to use the diagram of the linesand transversal from Try It. After students have used angle relationships tofind m/BCF, have them measure /BCF with aprotractor to verify their answer. (Students mayneed to extend the lines in the figure in order tomeasure the angles accurately.) Instruct students to use angle relationships tofind m/ACB. Ask: What is m/ACB? [110 ] Have them measure /ACB with a protractor toverify their answer. Instruct students to use angle relationships tofind m/EFC. Ask: What is m/EFC? [125 ] Have students measure /EFC
MATERIALS DIFFERENTIATION Curriculum Associates, LLC Copying is not permitted. LESSON 6 Describe Angle Relationships 115b LESSON 6 Overview Pacing Guide Items marked with are available on the Teacher Toolbox. SESSION 1 Explore Angle Relationships(35–50 min) Start (5 min) Try It (5–10 min) Discuss It (10–15 min) Connect It (10–15 min)
Before measuring an angle, it is helpful to estimate the measure by using benchmarks, such as a right angle and a straight angle. For example, to estimate the measure of the blue angle below, compare it to a right angle and to a straight angle. 90 angle 180 angle A right angle has a measure of 90
An angle is said to be an obtuse angle if it is greater than 90 but is less than 120 . Straight Angle An angle whose measure is 180 is a straight angle. This angle got its name as it forms a straight line. AOB is a straight angle. Reflex Angle A reflex angle
Once the measure of an angle is known, the angle can be classified as one of three types of angles. These types are defined in relation to a right angle. Types of Angles A right angle m A 90 acute angle 0 m A 90 A obtuse angle 90 m A 180 A Angle Measure
Aug 18, 2015 · Measure and classify angles. Identify and use congruent angles and the bisector of an angle. Discover relationships between special pair of angles. Vocabulary Degree, ray, angle, sides, vertex, interior, exterior, right angle, acute angle, obtuse angle, angle bisector. Adjacent angles, linear pair angles, vertical ang
Vertical Angles: Two angles whose sides form two pairs of opposite rays. Acute Angle: An angle less than 90 degrees. Obtuse Angle: An angle greater than 90 degrees. Right Angle: An angle equal to 90 degrees. Straight Angle: An angle equal to 180 degrees. Midpoint: The point that divide
Prove few of the basic trigonometric identities. BACKGROUND: Reference Angle: For any given angle, its reference angle is the corresponding acute version of that angle. The reference angle is the smallest angle between the terminal side and the X-axis. Figure 1 shows the concept of a reference angle.
2.-Taking P dot on r as the vertex, indicate the resulting angle for the angle subtraction: A - B. 3.-Taking P dot on r as the vertex, indicate the resulting angle for the following operation: A B - C. 4.-Draw the angle bisector of the given angle. 1.-Taking P dot on r as the vertex, indicate the resulting angle for the angle addition: A B. P r A B A B C A B
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