Download chapter 5-2 - ASB-ScienceandMath

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Study Guide and Intervention
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lnequalities and Triangles
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When the sides of triangles
not congruent, there is a relationship between the sides and
For any real numbers o and b, either a < b,s : b, or a > b.
The Exterior Angle Theorem can be used to prove this inequality involving an exterior angle
o Ifone side ofa triangle is longer than another side, then the
lf an angle is an exterior angle of a
triangle. then its measure is greater than
lnequalityThsrem I the measure of erther of its cotresponding
remote interior angles.
Angle-Side Relationships
ffi
nzl.
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Listanangles of LElGwhosemeasulesare
The measure ofan exterior angie is greater than the measure of
either remote interior angle. So mL\ < mLl and, mL4 < mLI.
List all aFgles that satisfy the stated condition.
1.
all angles whose measures are less than
3, all ang.les whose measures are less than
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mLB L|, L5
mz_l L7
5. all angles whose measures are less than
7.
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L4,L5,L7?LNPR
10.
mZl
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all angles whose measures are greater than mZ4
L1, L8, LOPN(_LROQ,/)
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LC, LB, LA
(tl
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Detemine the relationship between the measures of the
4.LR,LRUS mLR< nLRUS
5.
L1, L7, LTIIV
LT, LUST
mLT> nLUST
6.LUVS,LR MLUVS> MLR
Detemine the relationship between the lengths of the
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given sides.
7.AC,BC AC> BC
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8.BC,DB BC> DB
9.AC,DB AC> DB
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Glencoe Geometty
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CB,AB,AC
given angles.
Exercises
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LT, LR, LS
thm mt_4 L2, L3
9. all angles whose measures are less than
List the sides in order
from shortest to longest.
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mL2 L4
all angles whose measures are greater than mLb
8. all angles whose measures are less
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mL7 L1, L3, L5, L6, L TUV
6. all angles whose measures are greater than
r.
4/i\
Exercises
r Ac > AB, then mLB > mLc.
n mLA> mLc.then Bc> AB.
List the angles or sides in order from least to greatest measure.
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Exercises 1-2
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order I W
LT, LR, LS
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mL7 L5, LG
4. all angles whose measures are greater than
\1\2--\3\\.
A
angie opposite the longer side has a greater measure than the
angle opposite the shorter side.
Ifone angle ofa triangle has a greater measure than another
angle, then the side opposite the greater angle is longer than
the side opposite the lesser angle.
s
ocm/\766
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are
triangles.
List the angles in
f,ffiM*
from least to greatest measure.
LN
mLl L3, L4
2. all angles whose measures are greater than
angles ofthe
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mL1 > mLA. mL1 > mLB
less than
pontinued)
lnequalities and Triangles
Propcrtics of incqualitics, including the Transitive, Addition,
Subtraction, Multiplication, and Division Properties of Ineqirality, can be used. with
measures of angles and segments. There is also a Comparison property of Inequality.
Angle
PEHIoD
5-ff Study Guide and Intervention
Angle lnequalities
Exterior
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Glencoe Geometry
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5-2* Practice
5-2d Skills Practice
lnequalities and Triangles
lnequalities and Triangles
Detemine which angle has tbe gyeatest measure.
I. LI, L3, L4
L1
2. L4, L5, L7
L6
4. L5, L6, L8
3. L2, L3,
L6
.
L4
A\
l,/2 4\6 7\8
Determine which angle has the greatest neasure.
I. LT, L3, L4
L1
>
3. L2, L3, L7
L7
L8
Use the
Exterior Angle Inequality Theorem to list all
angles that satisfy the stated condition.
L2, L3, L4, L5, L7, L8
/,/\.s\o Nq
r,/ :/
Exterior Angle Inequality Theorem to list
5. all angles whose measures are less than mZ1
L3, L4, L5, L7, L8
,
J
L5, L7, L8
mzg
L2, L4, L6, L7
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7. all angles whose measures are greater than mZZ
L1, L3, L5, L9
all angles whose measures are greater than mL1
L1, L3
ct)
4. L7, L8, LIO
L10
6. all angles whose measures are less than m1-3
6. all angles whose measures are less than
7.
2. L4, L8, L9
L4
all angles that satisfu the stated condition.
Use the
5, all angles whose measures are less than mZ1
laveraee;
-v,
8. all angles whose measures are greater Lhan mL2
L6, L9
8. all angles whose measures are greater than mZ8
L1, L9, L5
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Detemine the relationship between the measures
of the given angles.
9.nLQRW,nLRWQ
Determine the relationship between the measures of
the given angles.
9.
mIABD,wLBAD
nLABD> nLBAD
l1..ryLBCD,mLCDB
nLBCD> nLCDB
lO.mlADB,qLBAD
l1..
nLADB < nLBAD
nLRST,wLTRS
12.mLCBD,mLCDB
DH>
given sides.
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LM<LP
EG<FG
I4. MP,MN
MP> MN
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MN<NP
16, MP,
< LQRW
14. DE, DG
DE<DG
GH
T5.EG,FC;
L6,
LP
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,UqAVZ.
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DE> EG
between the trees the greatest? 2
MP<LP
K-7
DE,EG
17. SPORTS The figure shows the position ofthree trees on one
part of a FrisbeelNl course. At which tree position is the angle
(D
L5.MN,NP
MLWOR
of the given sides,
Detemine the relationship between the lengths of the
LM, LP
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12.nLWQR,nLQRW
Detemine the relationship between the lengths
nLCBD> nLCDB
=
('l
MLRTW < LTWR
MLRST> LTRS
13.DH,GH
T3.
IO.wLRTW,qLTWR
MLQRW < LRWQ
a2
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Glencoe Geometty
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Glencoe Geometry
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S;ffi Reading to Learn Mathematics
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Pre-Activity How can you
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Construction Problem
t'ell which comer is bigger?
Read the introduction to Lesson 5-2 at the top ol page 247 in your textbook.
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Enrichment
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Inequalities and Triangles
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Which side ofthe patio is opposite the largest corner?
Which side ofthe patio is opposite the smallest corner?
the 51-fOOt Side
the 4S-foot side
The diagram below shows segment AB adjacent to a closed region. The
problem requires that you construct another segment Xf to the right of the
closed region such that points A, B, X, and Y are collinear. You are not allowed
to touch or cross the closed region with your compass or straightedge.
Reading the Lesson
1. Name the property
ofinequality that is illustrated by each ofthe following.
a lfr > 8 and 8 > y, then r > y. Transitive Property
b. lf x<y,thenr - 7.5<y * ?.5. Subtraction Property
c. lf x > y, then -3r < -3y. Multiplication Property
d.
Ifris
anyrealnumber,r > 0,r = 0,orr
2. Use the definition of inequality to
wite
a.2O>122O:12+8
c.8>-28:-2+10
e. -11 > *L2
-1'l = -12 + 1
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0. COmpafisOn PrOpefty
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an equatioz that shows that each inequality is true.
b.101>99101:99+2
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d-7>-77:-7+14
f. -30 > -45 -30 : -45 + 15
nLIJK : 45 and mLH > mLI.
a. Arrange the following angles in order from largest to
3. In theffigure,
sma]]est:
LI, LIJK, LH, LIJH LIJH, LIJK, LH, LI
H
b. Arrange the sides of LHIJ in order from shortest to longest.
Follow these instructions to construct a segment XY so that it is
collinear with segment AIl.
HJ, IJ, HI
c. Is AIIIJ an acute, right, or obtuse trianglg? Explain your reasoning.
Obtuse; sample answerr LUHis obtuse because
nLIJH : 180 - nLIJK = 135. Therefore, AHIJ is obtuse because it
has an obtuse angle.
1. Construct the perpenilicular bisector of AB-. Label the
midpoint
as
point
C,
and the line as m
to
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(Jl
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2. Mark two points P and Q on line m that lie well above the closed region.
d. Is AI11"I scalene, isosceles, or equilateral? Explain your reasoning.
Scalene; sample answer: the three angles ol LHIJ all have different
measures, so the sides opposite them must have different lengths.
Construct the perpendicular bisector z ofPQ. Label the intersection of
rn and z as point D.
lines
line nthatlie well to,the right of the closed
region. Construct the perpendicular bisector (offiS. Label the
intersection of lines a and (as point E.
3. Mark points 7i and S on
Helping You Remember
it to a theorem you
learned earlier. Explain how the Exterior Algle Inequality Theoren is related to the
Exterior Angle Theorem, and why the Exterior Angle Inequality Theorem must be true
the Exterior Angle Theorem is true.
4. A good way to remember a new geometric theorem is to relate
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Sample answer: The Exterior Angle Theorem says that the measure of an
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Glencoe/McGraw-Hill
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Glencoe Geometty
(so ihat X is below line
z
md
so
that EX is
(and on opposite sides ofX, so that X? and
-YV are congruent. Construct the perpendicular bisector { of 7V. CalI the
point where the line { hits the boundary of the closed region point Y. XY
5. Mark points ? and V on line
exterior angle of a triangle is equal to the sum ol the measures of the
two remote interior angles, while the Exterior Angle lnequality Theorem
says that the measure of an exterior angle is greater than the measure of
either remote interior angle. lf a number is equal to the sum of two
positive numbers, it must be greater than each of those two numbers.
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4. Mark point X m line
congruent to DC.
if
conesponds to the new road.
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Glencoe/Mccraw-Hill
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Glencoe Geometry
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