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T616
Mathematics Success – Grade 8
LESSON 24: Angle Relationships – Part 1
[OBJECTIVE]
The student will explore angle relationships within triangles.
[PREREQUISITE SKILLS]
angles, congruency, supplementary angles, adjacent angles
[MATERIALS]
Student pages S307–S314
Plain Paper
Ruler or Straight-edge
Protractor
Tape or Glue
Scissors
[ESSENTIAL QUESTIONS]
1. What angle relationship did we discover about the interior angles of a triangle?
2. Explain the relationship between an exterior angle of a triangle and the two nonadjacent interior angles.
3. What is the relationship between an exterior angle of a triangle and the adjacent
angle?
[WORDS FOR WORD WALL]
interior angles, exterior angles, supplementary angles, adjacent interior angle,
non-adjacent interior 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) S307 (Answers on T624.)
Have students turn to S307 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 group. {Graphic Organizer, Pictorial Representation}
[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
T617
LESSON 24: Angle Relationships – Part 1
SOLVE Problem
(WG, GP) S308 (Answers on T625.)
HavestudentsturntoS308intheirbooks.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 and determine the relationships
between angles. 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 – Interior Angles of Triangles
(M, GP, CP, WG) S308 (Answers on T625.)
M, CP, GP, WG:
After students complete the S Step of the SOLVE problem,
distribute two pieces of paper to each student to complete
the next activity. Prior to beginning the activity, be sure to
identify Partners A and B for activities. {Verbal Description.
Concrete Representation, Graphic Organizer}
MODELING
Discovery Activity – Interior Angles of Triangles
Step 1: Have each student draw three points on the piece of paper. What shape
would we create if we connected the three points on the paper? (a
triangle)
*Teacher Note: Have the students place the points far enough apart that they
will have a large enough triangle to cut. While we want very different triangles for
each student, it’s important that the triangle is large enough for the activity.
• N
ext,havestudentsuseastraight-edgetoconnectallthreepoints,
creating a triangle.
• A
fterstudentshavefinisheddrawingtheirtriangles,takeamomentto
have the students hold their drawings up so that the rest of the class
can see the triangle. Students may do this all at the same time.
• W
hatdoyounoticeaboutthetrianglesthatwehavedrawn?(Students
may have triangles that look somewhat similar to their own or they
may see some that look very different. The only true similarity is that
each students’ drawing is a triangle.)
• Modelhowtotearaparttheanglesofthetriangle.
• Wehavetwooptionsfortearingthetriangle.Seetheoptionsbelow.
1. Tear off the points of the triangle such that you tear off three
angles.
T618
Mathematics Success – Grade 8
LESSON 24: Angle Relationships – Part 1
2. Tear so that three pieces of the triangle are created
• E
itherofthewaysdisplayedwillworkforthisactivity.Thepointis
that we will have the three angles of the triangle available to make
the discovery.
• Next,havestudentsplacethethreeanglesonthesecondpieceof
blank paper that was distributed.
• Havestudentslineuptheanglessothatthesidesofthetriangle
are adjacent, as displayed below. The solid black lines represent
the sides of triangle that aren’t torn and the dashed lines represent
where the triangle was torn.
• P
artnerA,describewhatyounoticeabouttheanglesofthetriangle
when laid next to each other. (Together they create a straight line.)
Record.
• Havestudentsuseaprotractororstraight-edgetoseethatit
creates a line of 180 degrees.
• PartnerA,useaprotractortomeasureeachofthethreeanglesof
your triangle, while Partner B does the same for his or her angles.
Record them in the chart at the bottom of S308. What is the sum of
the measures of the angles? (180 degrees) Record.
• Whatdoyounoticeaboutyouranglesandyourpartner’sangles?
(They are different measurements.)
• Whatdoyounoticeaboutthesumoftheanglesforyouandyour
partner? (They are both 180 degrees.)
• PartnerB,whatcanweconcludeaboutthesumofthe(interior), or
inside, angles of a triangle? The sum is always (180) degrees.
Mathematics Success – Grade 8
T619
LESSON 24: Angle Relationships – Part 1
Finding Missing Angles
M, GP, CP, WG:
(M, GP, CP, IP, WG) S309 (Answers on T626.)
Using the knowledge that the sum of the interior angles
of a triangle is always equal to 180 degrees, students will
solve for the missing angles in triangles that are provided.
Be sure that students know their designation as Partner A
or Partner B. {Verbal Description, Graphic Organizer, Pictorial
Representation}
MODELING
Finding Missing Angles
Step 1: Direct students’ attention to Question 1.
• PartnerA,whatdoweknowabouttheanglesinthegiventriangle?
(One angle measures 140 degrees and a second angle measures 27
degrees, while the third angle is labeled as Angle a.)
• Havestudentsdiscusshowtheymaybeabletousetheinformation
given to determine the third angle measurement.
• PartnerB,whatdoweknowaboutthesumoftheinterioranglesofa
triangle? (The sum of the interior angles of all triangles is 180 degrees.)
• PartnerA,explainhowwecanfindthemissingangleandjustifyyour
answer. (Find the value that when added to 140 and 27 will equal 180
degrees.)
• PartnerB,whydowewantthetotaltobe180degrees?(Thesumof
the interior angles of all triangles is 180 degrees.)
• PartnerA,whatoperationcanweuse?(subtraction)
• PartnerB,explainhowwecanfindthemissingangleusingsubtraction.
(180 – 140 – 27) Record.
• PartnerA,whatisthedifferenceafterthesubtraction?(13degrees.)
• PartnerB,whatisthemeasureofAnglea? (13 degrees) Record.
Step 2: Direct students’ attention to Problem 3.
• PartnerB,explainhowthisproblemisdifferentfromProblem1.(We
are only given one angle measurement.)
• PartnerA,whatdothemarkingsontheothertwoanglesmean?(The
arcs tell us that those two angles are congruent.)
• Havestudentpairsdiscusshowtheycanusetheinformationtheyare
giventofindthemeasureofthemissingangles.
• PartnerB,explainthefirststeptofindthemeasurementsofthesetwo
angles.(Subtractthe98degreesfrom180tofindthemeasurement
of the sum of the other two angles.)
• Partner A, what is the sum of the other two angles? Defend your
answer. (180 – 98 = 82 degrees.) Record.
T620
Mathematics Success – Grade 8
LESSON 24: Angle Relationships – Part 1
• P
artnerA,ifthetwoanglesleftoverarecongruent,howcanwefind
what each of their measures are? Justify your answer. (Divide 82 by
2 to get an angle measure of 41 degrees.) Record.
• PartnerB,whatarethemeasuresofeachoftheremainingangles?
(Each angle has a measure of 41 degrees.) Record.
Step 3: Direct students’ attention to Problem 7.
• PartnerA,howisthisproblemdifferentthantheothers?(Itdoesnot
have a drawing, but only given angle measures.)
• Partner B, explain how we can solve this problem? (Use the same
strategyofsubtractingthegivenanglesfrom180degreestofindthe
remaining angle.)
• PartnerA,whataretheremainingdegreesafterwesubtracttheknown
angles? (180 – 75 – 25 = 80 degrees) Record.
• PartnerB,whatisthemeasureofAngleC? Justify your answer. (80
degrees)
IP, CP, WG:
Have students work in student pairs to complete the
rest of page by completing Problems 2 – 6 and 8 – 10.
After students have completed the problems, review the
answers as a whole group. {Verbal Description, Graphic
Organizer, Pictorial Representation}
Introduction to Exterior Angles
(M, GP, CP, IP, WG) S310, S311 (Answers on T627, T628.)
GP, CP, WG, M:
Students will explore three of the same triangles with three
different exterior angles and explore the characteristics
and relationship between the three. Be sure students know
their roles as Partner A or Partner B. {Verbal Description,
Pictorial Representation, Graphic Organizer}
MODELING
Introduction to Exterior Angles
Step 1: On pages S308 and S309 we worked with angles on the inside of the
triangle or (interior) angles. Record.
• Now we are going to look at the relationships of the angles on the
outside of the triangle.
• Partner A, can you identify the term for these angles? (exterior)
Record.
• Takealookatthefirsttriangle.StartyourpencilatPointA and trace
until you get to Point B.Now,withastraight-edgeorruler,continue
the line segment by extending AB past Point B.
Mathematics Success – Grade 8
T621
LESSON 24: Angle Relationships – Part 1
• P
artner A, explain what we created when we extended the line
segment. (We created a new angle.)
• PartnerB,describewhattypeofangleisontheoutsideofthefigure.
Defend your thinking. (an exterior angle, because it is outside the
triangle)
• PartnerA,whatdoyounoticeaboutAngleB and the new angle that is
created with the extension? (The two angles create a straight line.)
• Partner B, what is the total measure of Angle B and the new angle
measure when added together? Justify your answer. (180° because it
is a straight line.)
• PartnerB,identifyandexplaintherelationshipbetweentheadjacent
interior angle and the exterior angle. (The adjacent interior angle and
the exterior angle are supplementary. If one of the two is known, the
other one can be found by subtracting the known angle from 180°.)
Step 2: PartnerB,howcanwefindthemeasureofthenewangle?(Subtractthe
measure of Angle B from 180°.)
• PartnerA,whatisthemeasureoftheexterioranglethatiscreated?
(154°) Record.
• Let’sstudythetriangle’sinterioranglesandthenewexteriorangle.
Do you see any relationship between the interior and exterior angles?
(Students may begin to identify the connection between the two nonadjacent angles and the exterior angles.)
• PartnerB,Whatisthesumoftheinterioranglesthatarenotadjacent
to the exterior angle? (23 + 131 = 154°) Record.
• Whatdoyounoticeaboutthesumofthenon-adjacent interior angles
and the exterior angle? (They have the same measure of 154°.) Record.
Step 3: Direct students to the second triangle on S310.
• Takealookatthesecondtriangle.StartyourpencilatPointB and
trace until you get to Point C. Now, with a straight-edge or ruler,
continue the line segment by extending past Point C.
• PartnerA,whatdoyounoticeaboutAngleC and the new angle that was
created with the extension? (The two angles form a straight line.)
• Partner B, what is the total measure of Angle C and the new angle
measure when added together? Justify your answer. (180° because it
is a straight line.)
Step 4: PartnerA,howcanwefindthemeasureofthenewangle?(Subtractthe
measure of Angle C from 180°.)
• PartnerB,whatisthemeasureoftheexterioranglethatiscreated?
(157°) Record.
• PartnerA,identifythesumoftheinterioranglesthatarenotadjacentto
the exterior angle? Explain your thinking (26 + 131 = 157°) Record.
T622
Mathematics Success – Grade 8
LESSON 24: Angle Relationships – Part 1
• W
hatdoyounoticeaboutthesumofthenon-adjacentinteriorangles
and the exterior angle? (They have the same measure of 157°.) Record.
Step 5: Have students complete the process of determining the measure of the
exterior angle for Triangle 3 with their partners and then review the
answers as a whole group.
Step 6: Direct students to the top of S311.
• Partner A, from our discoveries on the previous page, what do we
know about the angles that are not adjacent to the exterior angle?
(The sum of the non-adjacent interior angles is equal to the exterior
angle.) Record.
• Partner B, explain the relationship between the adjacent angle and
the exterior angle. (The adjacent angle and the exterior angle are
supplementary. If one of the two is known, the other one can be found
by subtracting the known angle from 180°.) Record.
Step 7: Direct students to Question 3.
• PartnerA,whatistheproblemaskingustofind?(themissingangle
measures)
• Partner B, using the strategy from the previous page, explain how
wecanfindthemeasureofAngled. (The exterior angle is equal to
the sum of the non-adjacent angles, so we can add the non-adjacent
angles).
• PartnerA,whatwillbethesumthatequalsAngled? (91 + 31 = 122°)
Record.
• Partner B, how can we find the measure of Angle a? Defend your
answer. (Angle d and Angle a are supplementary angles, therefore we
can subtract the measure of Angle d from 180°.)
• PartnerA,whatisthemeasureofAnglea? (180 – 122 = 58°) Record.
• Partner B, is there another way we could have found the measure
of Angle a if we wouldn’t have known the value of the Angle d? (We
could subtract the other two interior angles from 180° because the
sum of interior angles has to be 180°.)
IP, CP, WG:
Students will complete Problems 4 – 6 in student pairs to
practice identifying the missing angle measures. They will
use their knowledge that the sum of the interior angles of
a triangle will total to 180° as well as the two connections
regarding interior and exterior angles that they concluded
at the top of S311. After students have completed the
problems, review the answers as a whole group. {Verbal
Description, Graphic Organizer, Pictorial Representation}