lesson 25: angle relationships – part 2 objectiventnmath.kemsmath.com/level h lesson notes/grade...

Mathematics Success – Grade 8T634

LESSON 25: Angle Relationships – Part 2

[OBJECTIVE]The student will explore angle relationships created by parallel lines cut by a transversal in order to solve real world problems.

[PREREQUISITE SKILLS]congruency, vertical angles, supplementary angles, adjacent angles

[MATERIALS]Student pages S315–S325RulerProtractorSticky NotesColored Pencils

[ESSENTIAL QUESTIONS]1. What types of angle relationships are created when we use a transversal to cut

through parallel lines?2. If parallel lines are cut by only one transversal, how many different measures of

angles are created?3. Where can parallel lines cut by a transversal be seen in real life situations?

[WORDS FOR WORD WALL]parallel lines, transversal, corresponding angles, alternate interior angles, alternate exterior angles, vertical angles

[GROUPING]Cooperative Pairs (CP), Whole Group (WG), Individual (I)

*For Cooperative Pairs (CP) activities, assign the roles of Partner A or Partner B to students. This allows each student to be responsible for designated tasks within the lesson.

[LEVELS OF TEACHER SUPPORT]Modeling (M), Guided Practice (GP), Independent Practice (IP)

[MULTIPLE REPRESENTATIONS]SOLVE, Verbal Description, Pictorial Representation, Concrete Representation, Graphic Organizer

[WARM-UP] (IP, I, WG) S315 (Answers on T646.)Have students turn to S315 in their books to begin the Warm-Up. Students will use knowledge of angles to complete the assignment. Monitor students to see if any of them need help during the Warm-Up. After students have completed the Warm-Up, review the solutions as a whole group. {Graphic Organizer, Pictorial Representation, Verbal Description}

[HOMEWORK]Take time to go over the homework from the previous night.

[LESSON] [1 – 2 Days (1 day = 80 minutes) – M, GP, WG, CP, IP]

Mathematics Success – Grade 8 T635


MODELING

Discovery Activity – Angle Relationships

Step 1: Direct students to the blank space on the bottom of S316. • PartnerA,placetherulerinthemiddleoftheblankspaceandholdit

there so that the ruler is creating a horizontal line. • PartnerB,whilePartnerAholds the ruler,useyourpenciland the

ruler to draw a line above the ruler and below the ruler in your own book. Note that Partner A should keep the ruler in the same position while all lines are being drawn. Switch roles and repeat this step so Partner A also has a set of lines in his or her book. Record the lines on the paper.

• Adddirectionalarrowstothelines.PartnerA,whatdothesearrowsmean? (They mean that the line continues on forever.)

• Be sure that you are completing this activity as a model for thestudents as well. You will use your model as a guide for students.

• Labelthetopline“a”andthebottomline“b.” It doesn’t matter which oneis“a”andwhichoneis“b.” Having them in the same place as your model makes it easier to refer to when completing the activity.

• PartnerB,describewhatwehavecreatedusingtherulerasourguide.(a set of parallel lines)

• PartnerA,wheremightweseearealworldexampleofparallellines?(possible answers: railroad tracks, top and bottom edge of math book, left and right edge of desk)

SOLVE Problem (WG, GP) S316 (Answers on T647.)

HavestudentsturntoS316intheirbooks.ThefirstproblemisaSOLVEproblem.You are only going to complete the S step with students at this point. Tell students that during the lesson they will learn how to explore the relationships between angles that are formed with parallel lines and transversals. They will use this knowledge to complete this SOLVE problem at the end of the lesson. {SOLVE, Verbal Description, Graphic Organizer, Pictorial Representation}

Discovery Activity – Angle Relationships (M, GP, CP, WG) S316, S317 (Answers on T647, T648.)

M, GP, CP, WG: After students complete the S Step of the SOLVE problem, distribute rulers and protractors to each pair of students. Also, be sure to provide each student with two sticky notes. Students will be discovering the angle relationships that are created with parallel lines cut by a transversal. Be sure that students know their responsibilities as Partners A and B for activities. {Verbal Description, Pictorial Representation, Graphic Organizer}



• Havestudentpairsdiscussapossibledefinitionofparallellinesbasedon the real world examples. (Parallel lines are a pair of lines that never intersect.)

Step 2: Model for students the symbol that we use to identify a pair of lines as parallelbydrawingthekeythatsays“a || b.”) Have students denote parallel lines on their drawing.

• Partner B, place the ruler so that it rises from left to right and isintersecting both of the parallel lines on the paper. Partner A, draw the intersectingline.Labelthisline“t.” Be sure that you add arrows at the end of the line.

Step 3: Partner A, how can we describe the line we just drew? (It intersects or crosses the two parallel lines.)

• Havestudentsdiscusswhyweusedtheletter“t” to represent the line that intersects the parallel lines.

• Let’sdiscussanotherwordwecoulduseforalinethatcrossesasetof parallel lines. (a transversal which means – a line that passes through two lines in the same plane)

• Havestudentslabeleachoftheanglesontheirdrawingthesameasyours so that when you refer to an angle, you are referring to the same angle as the students. Record angle names.

Step 4: Each student will need to use a protractor and measure Angle 1. Model the measure of Angle 1 for students. Record the measure in the section with Angle 1.

• Takea lookatAngle1andAngle2.PartnerA,whatdoyounoticeabout Angle 1 and Angle 2? (They form a straight line.)

• PartnerB,whatissumofthedegreesofAngles1and2?Justifyyour answer. (180° because they create a straight line.)

• PartnerA,ifthetotalofthetwoanglesis180°,howcanwefindthemeasure of Angle 2? (Subtract my measure for Angle 1 from 180°.)

• Have students subtract Angle 1 from 180° to find themeasure indegrees of Angle 2. Then, have students write the measure in the section with Angle 2.

Step 5: Ask students to retrieve their sticky notes. • AllstudentsshouldplaceastickynoteovertopofAngle1.Usinga

pencil, each student should trace Angle 1 on the sticky note. Encourage students to use the ruler or protractor as the straight edge for tracing the angle. Have students label Angle 1 on the sticky note.



• HavestudentsrepeatthetracingprocessforAngle2,usingthesecondsticky note. Students should label Angle 2 on this sticky note.

• Usingthetwoanglesthatyoutraced,explorethemeasuresofAngles3 and 4. Give students a moment to manipulate the traced angles to see that the measures are identical for Angles 3 and 4.

• PartnerB,doyounoticeanyconnectionswithAngle3?(Angle3isthe same measure as Angle 2.) Have students go back to the original drawing and record the measure of Angle 3 so that it matches Angle 2.

Step 6: Partner A, what relationship do Angles 2 and 3 share that allows them to have the same measure? (They are vertical angles.)

• PartnerB,whatisadescriptionofverticalangles?(anglesoppositeofeach other when two lines intersect)

• Partner A, what do we know about vertical angles? (They arecongruent.)

• PartnerB,whatisanotherwaywecouldhaveidentifiedthemeasureof Angle 3, not knowing the measure of Angle 2? (Angles 1 and 3 are supplementary because they create a straight line, therefore, we couldagainsubtractthemeasureofAngle1from180°tofindthemeasure of Angle 3.)

Step 7: Partner A, do you notice any connections with Angle 4? (Angle 4 is the same measure as Angle 1.) Record the measure of Angle 4 on the original drawing.

• PartnerB,explainwhatrelationshipAngles1and4share.(Theyarevertical angles.)

• Partner A,what relationship does Angle 4 sharewith Angle 2 andwith Angle 3? (They are supplementary angles, therefore each pair of angles should total 180°.)

Step 8: Direct students to S317. • Whatisthetitleofthisgraphicorganizer?(TypesofAngles) • Inthetopleftbox,whatisthefirsttypeofangle?(VerticalAngles) • Partner B, describe vertical angles. (angles opposite of each other

when two lines intersect) Record. • Askstudentstonamepairsofvertical angles. (Angle 1 and Angle 4,

Angle 2 and Angle 3) Record. • Askstudentstolookattheoriginaldrawing. • PartnerA,canyouidentifyanyotherpairsofanglesonouroriginal

drawing that meet the description of vertical angles? (Yes, Angle 5 and Angle 8, as well as Angle 6 and Angle 7) Record the angle congruenciesintheboxnextto“CongruentAngles”onS317.

*Teacher Note: Review the symbol for congruent (≅) as needed.



Step 9: Direct students back to S316. • Usingyourstickynotesagain,takeamomenttoexplorethemeasures

of Angles 5 – 8. • PartnerA,whatdoyounoticeaboutAngle5?(Itisthesamemeasure

as Angle 1.) Record the measure of Angle 5 in the section. • PartnerB,whatdoyounoticeaboutthestickynoteifyoustartitat

Angle 1 and then line it up with Angle 5? (You do not need to rotate it, only slide it into position.)

• PartnerB,whatdoyounoticeaboutAngle6?(Itisthesamemeasureas Angle 2.) Record the measure of Angle 6 in the section.

• PartnerA,whatdoyounoticeaboutthestickynoteifyoustartitatAngle 2 and then line it up with Angle 6? (You do not need to rotate it, only slide it into position.)

• Explain to students thatwhenangles are on the same side of thetransversal, with the same measure, they are called (corresponding angles).

• PartnerA,howcanwedescribecorrespondingangles?(anglesonthesame side of the transversal, and same placement in relation to the parallel line) Record on S317.

• Partner B, what are some examples of congruent correspondingangles? (Angle 1 and Angle 5, Angle 2 and Angle 6, Angle 3 and Angle 7,Angle4andAngle8)Recordnextto“CongruentAngles.”

Step 10: Direct students back to S316. • PartnerA,howcanwefindthemeasureofAngle7?(Itissupplementary

to Angle 5, and it is also the same as Angle 6 because they are vertical angles.) Have students record the measure of Angle 7.

• PartnerB,howcanwefindthemeasureofAngle8?(Itissupplementaryto both Angle 7 and Angle 6, and it is also the same as Angle 5 because they are vertical angles.) Record the measure of Angle 8.

Step 11: Partner A, what do you notice about Angles 4 and 5? (They are congruent because they have the same measure.)

• Partner B, what do you notice about Angles 3 and 6? (They arecongruent because they have the same measure.)

• These angles share a relationship.What do you think we can callthese angles that are on opposite sides of the transversal and inside of the parallel lines? (alternate interior angles)



• PartnerA,howcanwedescribe theseangles? (Anglesonoppositesides of the transversal inside of the parallel lines) Record on S317.

• PartnerB,whatarethecongruentalternateinteriorangles?(Angle3and Angle 6, Angle 4 and Angle 5) Record.

Step 12: Direct students back to S316. • Partner B, what do you notice about Angles 1 and 8? (They are

congruent because they have the same measure.) • Partner A, what do you notice about Angles 2 and 7? (They are

congruent because they have the same measure.) • These angles share a relationship.What do you think we can call

these angles that are on opposite sides of the transversal and outside of the parallel lines? (alternate exterior angles)

• PartnerB,howcanwedescribe theseangles? (anglesonoppositesides of the transversal outside of the parallel lines) Record on S317.

• PartnerA,whatarethecongruentalternateexteriorangles?(Angle1and Angle 8, Angle 2 and Angle 7) Record.

*Teacher Note: A common challenge for students is identifying the interior and the exterior on a diagram. Ask students for suggestions of how they see the inside and outside of the parallel lines (i.e. inside/outside of a sandwich, inside/outside of the tracks, inside/outside the skis). It is also helpful to provide students with two lightly colored pencils and have them shade the interior one color and shade the exterior a second color – labeling both on the diagram.

• Whatdoyounoticeaboutthenumberofdifferentmeasuresofangleson the diagram we created on S316? (There are only two different measures.)

• Therefore,wecanuseallofourdifferentanglerelationshipstofindthe measures of the angles.

Horizontal Parallel Lines – Finding Missing Angles (M, GP, CP, IP, WG) S318 (Answers on T649.)

M, GP, CP, WG: Students will apply what they have discovered in the previous activity. Using the knowledge of the different angle relationships, students will label the different measures of the angles. Be sure that students know their responsibilities as Partner A and Partner B. {Verbal Description, Graphic Organizer, Pictorial Representation}


MODELING

Horizontal Parallel Lines – Finding Missing Angles

Step 1: Direct students’ attention to the top of S318.

• PartnerA,whatdoesthenotationinthebottomrightcornerofthebox for Question 1 represent? (It tells us that line m and line n are parallel to each other.)

• PartnerB,whatanglemeasurearewegiven?(Angle3=43°)

• RecordthemeasureofAngle3intheboxontheright.

• PartnerA,whatrelationshipdoAngle1andAngle3share?(Theyaresupplementary.)

• PartnerB,howcanwefindthemeasureofAngle1?(Subtract180–43=137°)RecordthemeasureofAngle1intheboxontheright.

• PartnerB,whatistherelationshipbetweenAngle1andAngle4,aswell as Angle 3 and Angle 2? (They are vertical angles, which means they are congruent.) Record the measures of Angle 2 and Angle 4 in the box on the right.

• PartnerA,howdoesAngle3comparetoAngle6?(Theyarecongruentbecause they are alternate interior angles.) Record 43° for Angle 6 in the box on the right.

• PartnerB,whatisanotherpairofalternateinteriorangles?(Angles4and 5)

• What can we record as the measure of Angle 5? (137° becausealternate interior angles are congruent.) Record.

• HowcanwefindthemeasuresforAngles7and8?(Wecouldusetheidea of vertical angles from Angles 5 and 6, supplementary angles from Angles 5 and 6, alternate exterior angles from Angles 1 and 2, or corresponding angles from 3 and 4) Record the measures of Angle 7 and Angle 8. (43° and 137°)


IP, CP, WG: Have students complete the rest of page by completing the angle question in Problem 2 and then completing the statements for Problems 3 – 5. Students can reference S317 to identify the different types of angles at the bottom of S318. After students have completed the activity, review the answers as a whole group. {Verbal Description, Graphic Organizer, Pictorial Representation}


MODELING

Vertical Parallel Lines – Finding Missing Angles

Step 1: Direct students to Question 1 on S319. • PartnerA,arelinesj and k parallel? (Yes) Record. • PartnerB,explainanddefendyouranswer.(Thetwoarrowsatthetop

of the parallel lines are symbols that the lines are parallel.) Record. • PartnerB,howisthisdiagramdifferentfromtheothersetsofparallel

lines that we have seen? (In this case, the parallel lines are vertical rather than horizontal.) Record.

• PartnerA,whereare theexteriorangles located?(Theyare to theright of line k and the to the left of line j.) Record.

• Let’s use our knowledge of angle relationships to find themissingangles and then complete the table at the bottom of the page.

Step 2: Partner B, what angle measure are we given? (Angle 7 is 133°) Record this measure in the box to the right.

• PartnerA,whatisthemeasureofAngle6?(47°)Defendyouranswer.(It is supplementary with Angle 7, therefore we can subtract 180 – 133tofindthedegreesforAngle6.)

• Partner B, how can we identify themeasures of Angles 5 and 8?Explain your thinking. (Angle 5 is the same as Angle 7 and Angle 6 is the same as Angle 8 because they are vertical angles.) Record the measures for Angles 5 and 8 in the box to the right.

• PartnerA,whataretheotherpairsofverticalangles?(Angle2andAngle 4, as well as Angle 1 and Angle 3) Record all pairs of vertical angles in the table at the bottom of the page.

• PartnerA,howdoesAngle3relatetoAngle7?Justifyyouranswer.(They are corresponding angles so they are congruent.) Record the measure of Angle 3 in the box to the right.

• Whataresomeothercorrespondingangles?(Angle6andAngle2,Angle 8 and Angle 4, Angle 5 and Angle 1.) Record these pairs in the table at the bottom of the page. Using corresponding angles, we are able to identify all of the missing measures in the box to the right. Have partners complete the box.

Vertical Parallel Lines – Finding Missing Angles (M, GP, CP, IP, WG) S319, S320 (Answers on T650, T651.)

GP, M, CP, WG: Students will look at parallel lines that are vertical and identify the different angle relationships that still exist when cut by a transversal. Be sure that students know their responsibilities as Partner A and Partner B. {Verbal Description, Pictorial Representation, Graphic Organizer}



MODELING

Multiple Transversals Problem Solving

Step 1: Direct students to the top of S321. • Useanglerelationshipsfromparallellinescutbyatransversalandthe

angles labeled below to informally prove that ∠1 + ∠2 + ∠3=180°. • Partner A, are the parallel lines horizontal or vertical? (They are

horizontal.) • PartnerB,whatisdifferentaboutthisdiagram?(Therearetwolines

that intersect the parallel lines and they form a triangle.) • PartnerA,arewegivenanyofthemeasuresofanyangles?(No.)

Step 2: Partner B, what do we know about ∠3, ∠4, and ∠5? Defend your answer. (They form a straight line, which means that their sum is 180°.) Record.

• PartnerB,wealsohavealternateinterioranglesinourdiagram.Whatangles are these? (Angle 3 and Angle 5, as well as Angle 4 and Angle 6) Record in the chart.

• PartnerA,thelastgroupisalternateexteriorangles.Whatanglesfallinto this category? (Angle 2 and Angle 8, as well as Angle 1 and Angle 7) Record.


IP, CP, WG: Students will complete Problems 1 – 4 on S320 to practice identifying missing angles with vertical parallel lines. Students should be able to complete this activity independently. Students may shade the interior and exterior angles with different colors to help them identify those angles. After student pairs have completed the activity, discuss the solutions as a whole group. {Verbal Description, Graphic Organizer, Pictorial Representation}

Multiple Transversals Problem Solving (M, GP, CP, WG) S321 (Answers on T652.)

GP, M, CP, WG: Students will look at parallel lines that are cut by two transversals. Students will use their knowledge of angle relationships to complete the informal proof on S321. Be sure that students know their responsibilities as Partner A and Partner B. {Verbal Description, Pictorial Representation}



• PartnerA,whatarewetryingtoprove?(WearetryingtoprovethatAngles 1, 2 and 3 also have a sum of 180°.)

• PartnerB,canweidentifyanytypeofrelationshipbetweenthesumof Angles 3, 4, and 5, and the sum of Angles 1, 2 and 3? (Yes, we can use angle relationships from parallel lines cut by a transversal.) Record.

• PartnerA,explainhowwecaninformallyprovethestatementcorrect.(If we can prove that Angles 1, 2 and 3 are equal to Angles 3, 4 and 5, then we can prove the statement correct.)

Step 3: Partner B, what angle relationships can we see from the diagram? (Angle 1 and Angle 5 are corresponding angles, which means that they are congruent.) Record.

• PartnerA,are thereanyotheranglerelationshipswecan identify?(Angle 2 and Angle 4 are alternate interior angles, which means that they are congruent.) Record.

• PartnerB,whatdotheserelationshipstellus?Justifyyouranswer.(IfAngle 1 is congruent to Angle 5 and Angle 2 is congruent to Angle 4, then ∠1 + ∠2 + ∠3=∠5 + ∠4 + ∠3=180°)Record.

• PartnerA,identifyanotheranglerelationshipwithtrianglesthatweknow that would verify this informal proof? (The sum of interior angles of any triangle is always 180°.) Record.

Problems with Parallel Lines Cut by Transversals (M, GP, CP, IP, WG) S322 (Answers on T653.)

GP, M, CP, WG: Students will take a look at parallel lines that are cut by transversals to solve real world problems. They will use their knowledge of angle relationships to draw conclusions and answer questions. Model your thinking and guide them through the problem solving process. Remember to include the thinking for SOLVE even though these are not written as a formal SOLVE problem. Be sure that students know their responsibilities as Partner A and Partner B. {Verbal Description, Pictorial Representation, Graphic Organizer, SOLVE}



MODELING

Problems with Parallel Lines Cut by Transversals

Step 1: Direct students to the top of S322.

• Readthefirstprobleminthetopofthebox. • PartnerA,what is theproblemaskingus tofind? (themeasureof

Angle B) • PartnerB,whatdoweknowaboutthestripes?Defendyouranswer.

(The two stripes are parallel lines.) Have students mark the stripes so they know these stripes are parallel.

• PartnerA,whatdoweknowaboutthelegsofthechair?(Theyarevertical parallel lines.) Again, have students mark these so they know they are parallel.

• PartnerB,whatinformationarewegiven?(Wehaveoneoftheanglesmeasuring 63°.)

• PartnerA, ifwewere touse the legsas theparallel linesand thetop stripe as the transversal, what angle can we identify? (We can identify the alternate interior angle from the top stripe, which will also measure 63°.) Have students mark this angle on the diagram.

• PartnerB,knowingthisangle,howcanwefindAngleB? Explain your thinking. (The new angle we just found is a corresponding angle to Angle B, therefore, they are congruent.)

• PartnerB,whatisthemeasureofAngleB? (It is also 63°.) Record. • Solution:Thealternateinteriorangleto63°canbelabeledonthetop

stripe. We know that the stripes are also parallel, so we can identify Angle B as a corresponding angle to the alternate interior angle. Therefore, Angle B is also 63°.

IP, CP, WG: Have students complete the second problem on S322. Allow students to work together and see if they can solve this problem on their own. The most challenging part of these problems is for students to solve the puzzle of how to connect the angles using what they are given. Regroup and review the solution to the problem. Encourage students to discuss different solutions because they may have used different angle relationships. {Verbal Description, Graphic Organizer, Pictorial Representation, SOLVE}

lesson 25: angle relationships – part 2 objectiventnmath.kemsmath.com/level h lesson notes/grade...

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