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You can use the Triangle Sum Conjectureto show why the Third Angle Conjecture
is true.Youll do this in Exercise15.
EXAMPLE
, Solution
In the figure at right, is ZC congruent
to ZE? Write a paragraphproof
explainingwhy.
o\A-------Ay,
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Yes.By the Third Angle Conjecture,
becauseLA and LB are congruentto LD
and,LE then LC must be congruent to ZE.
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y
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You couid also use the Triangle Sum Conjecture
to find that LD and ZF both measure50o.Since
they havethe samemeasure,they are congruent.
ERCISFS
Youwill need
chnologyUsing geometry software,construct a triangle. Use the software
) measurethe three anglesand calculatetheir sum. ,I.he
anglemeasurcs
changc,
Drag the verticesand describeyour observations.but thesumrenrains180".
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Geanetrytoftwote
for Exercise
1
funstructiontools
for Exercises
l0-13
, Repeat the process for a right triangle. illeasuresofthc acuteangles
-:ritnge,but studcntsmay notice or dcducc that their sum remirirrs90n.
,e Triangle Sum Conjecture to determine each lettered angle measure in Exercises
:u might find it helpful to copy the diagrams so you can write on them.
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> EXAMPLE
This example is good for
studentswho need more help in
understandingthe two conjectures of the investigation.
Remind studentsof the major
points of this lesson:The
Tiiangle Sum Conjecture says
that the sum of the measuresof
the anglesof any triangle is 180".
This conjecture can be proved
in a number of ways.One
consequenceis the Third Angle
Conjecture,which statesthat if
two anglesof one triangle are
congruent to two anglesof
another triangle, then their third
anglesalso have equal measure.
?. ,f,ir 110"
The exercisesfocus on the
Triangle Sum Conjecture.
ASSIGIIING
HOMETIIORK
Essential
6. Find the sum ofthe
7. Find the sum ofthe
measuresof the marked
measuresof the marked
angles.i/l 3' 360"- l8o"= 9oo' angles.t)
3' 180'- 180'= 360"
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2-16 evens
Performance
assessment 9
Portfolio
8
Journal
15
Group
1 - 1 3o d d s
Review
17-22
frtATtRtAtS
. Exercises
8 and9(I),optional
withtheExercises
) Helping
-r.. studentsare collectingdata and
- -:es from thesedata.
Assesstheir
r:,)tractor,a straightedge,
and patty
' ::rderstanding of acute triangle,
''.;Ltsureof an angle,and the Alter- i l's Coniecture.
Exercises
6,7 If students are
finding these exercisesdifiicult,
suggestthat they think about
subtraction or that they look at
the hints.
LESSON4.1 TriangleSum Conjecture 201
Exercises
8,9 These exercisesmay
be a bit intimidating at first.
You might do one in class,
with studentsworking in pairs
explaining why we know each
answer.This is another good
opportunity for think aloud pair
sharewhere students take turns
finding an answer and then
explaining it. As needed,point
out that some of the anglesare
in both a larger and a smaller
triangle. Remind studentsthat
practice makes perfect and that
it is much easierto practice in
pencil than in pen.
8.
?
..lt) 69"
?
9.m:
?
i0"
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?
50'
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?
82"
?
I 16'
?
93'
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?
28"
?
86'
--
?
31"
v-
7il"
I18"
50"
In Exercisesl0-12, use what you know to construct eachfigure.
Use only a compassand a straightedge.
Exercise
10 This construction
could also be done by
constructing the given angles
on the ends of a line segment;
then LM is the third angle
formed where the sides of the
anglesmeet.
1O. Construction
Given LA and LR of AARM, construct ZM.
11. ConstrudionIn AIEG, mLE : mLG. Given LL,
constructZG.
12. Construction
Given LA, LR, and side AE, construct AEAR.
13. Construction
RepeatExercisesl0-12 with patty-paper
constructions,
10.
14. In AMAS below, LM is a right angle.Let'scall the two
acute angles,LA and ZS, "wrong angles."Write a
paragraphproof or use algebrato show that "fwo wrongs
make a rightl'at least for anglesin a right triangle.
Frorn tl.rc'liiangleSum Oonjccturc
m L A I r t t L S I t t t L N l = l t l O ' .B c c a u s e
Zrtrl is a right ang|e,tnLhI = 90". lJ)'
sulrstitr.rtior.r,
rrZ A * rnLS t 90' = llt0"
l l v s u b t r a c t i o r r ,r t L A * n t L S = 9 t 1 " .
So two wrongs nrakea right!
12. First construct LE, using
the method used in Exercise10.
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13.
CHAPTER4 Discoveringand ProvingTriangleProperties
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lhad ... But loc Four wrongs s(
two wrongs lo lhs lourlh pow6l
thls tormula, do make a