1. No category

GUIDED
PRACTICE
THINK AND
DISCUSS
4-8
SEE EXAMPLE
p. 274
Vocabulary
Draw
isosceles
∠K as the
vertex
angle. Name
1.1. Explain
why each
of the
angles!JKL
in an with
equilateral
triangle
measures
60°. the legs, base,
and base angles of the triangle.
2. GET ORGANIZED Copy and complete the
/Àˆ>˜}i
organizer.
In each
box, draw
2. graphic
Surveying
To find
the distance
QRand
across a river, a surveyor locates three points Q,
1
KEYWORD: MG7 4-8
mark
a diagram
form,
each
of=
triangle.
R, and
S. QS = 41
andtype
m∠S
35°. The measure
of exterior
∠PQS = 70°. Draw a
µÕˆ>ÌiÀ>
µÕˆ>˜}Տ>À
diagram
and
can find QR.
explain
how you
Worksheet 4-­‐8 Exercises
KEYWORD: MG7 Parent
2
SEE EXAMPLE
p. 274
4-8
SEE EXAMPLE
p. 274
SEE EXAMPLE
Find each angle
measure.
GUIDED
PRACTICE
3. Vocabulary
m∠ECD
1.
Draw isosceles !JKL with ∠K4.asm∠K
the vertex angle. Name the legs, base,
and base angles of the triangle.
1
nÓÂ
ΣÂ
QR across a river, a surveyor locates three points Q,
2. SurveyingTo find the distance
R, and S. QS = 41 m, and m∠S = 35°. The measure of exterior ∠PQS = 70°. Draw a
Exercises
5. diagram
m∠X
and explain how you can find QR. 6. m∠A
2
SEE EXAMPLE
3
p. 275
1
SEE EXAMPLE
p. 274
SEE EXAMPLE
8 measure.
Find each angle
­xÌÊʣήÂ
9
­ÎÌÊÊήÂ
3. m∠ECD PRACTICE
GUIDED
p. 274
2
S E E E X A Mp.P274
LE 3
p. 275
SEE EXAMPLE 4
p. 275
4. m∠K
TAKS
SEE EXAMPLE 4
Skills Practice p. S11
Application Practice p. S31
0273_0279.indd 276
276
{ÝÂ
KEYWORD: MG7 Parent
each value. Draw isosceles !JKL with ∠K as the vertex angle. Name the legs, base,
Find
1. Vocabulary
nÓÂ
Σ
base
of the
7. and
y
8. x
, angles
- triangle.
£ÓÞÂ
2. Surveying To find the distance QR across a river, a surveyor locates three points Q,
5. m∠X
6. m∠A of exterior∠PQS = 70°. Draw a
R, and S. QS = 41<m, and m∠S = 35°. The measure
­£äÝÊÊÓä®Â
{ÝÂ
diagram 8
and /explain how9 you can find QR.
­xÌÊʣήÂ
­ÎÌÊÊήÂ
ÓÝÂ
each angle
measure.
9. BC
10. JK
Find
ÇÌÊÊ£x
3. m∠ECD
Find
each value.
7. y
ÞÊÊÓÎ
,
£ÓÞÂ
ÈÞÊÊÓ Î£Â
£äÌ
8. x
11. Given: !ABC is right isosceles. X is the
−− −− −−
midpoint of AC. AB $ BC
FindProve:
each angle
!AXBmeasure.
is isosceles.
13. m∠E
­£äÝÊÊÓä®Â
10. JK
8
8
Ó°{ʓˆ
­ä]Ê䮣äÌ
ÊÊÝÊ Ê
ÊÂ
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nä¨
{ä¨
Þ
/
8
,
12/7/05 1:59:55 PM
Ý
­ä]Êä®
-
­Ó>]Êä®
xÇÂ
­ÎÝÊÊ£ä®Â
16. m∠A
/
1
­ÈÞÊÊ£®Â
Find each value.
Ý
ÇÌÊÊ£x
­Ó>]Êä®
_0273_0279.indd 276
17. z
ÊÓä®Â
­£äÝÊ
Ý
­Ó>]Êä®
­ä]ÊÓ>®
{ÝÂ
ÇÌÊÊ£x
ÓÝ
Â
14. m∠TRU
Ó
nÓÂ
Þ
6. m∠A
­ä]ÊÓ>®
™ÈÂ
15. m∠F
11. Given: !ABC is right
isosceles. X is the
< −− −− −−
5. m∠X
/
midpoint of AC. AB $ BC
Chapter 4 Triangle Congruence
4. m∠K
plane is at A, an air-traffic controller
12
1
9. BC
in tower T measures the angle to
13–16
2
_0273_0279.indd 276
ÞÊAfter
ÊÓÎ the plane has
the plane as 40°.
17–20
3
traveled 2.4 mi to B, the angle to the
21
4
plane is
80°.
How
ÈÞÊ
ÊÓ can you find BT?
76
Chapter 4 Triangle Congruence
p. 275
KEYWORD: MG7 4-8
ÓÝÂ
8 is isosceles.9
9. BC
10. JK
Prove: !AXB
­xÌÊ
ʣή
­ÎÌÊ
Êή
Â
Â
ÞÊÊÓÎ
£äÌ
S E E E X A M P L E 3 Find each value.
­ä]Êä®
7. y
p. 275
8. x
, ÈÞÊÊÓ
£ÓÞÂ
276
Chapter 4 Triangle
Congruence AND PROBLEM SOLVING Þ
PRACTICE
Given: !ABC is right isosceles. X is the
S E E E X A M P L E 4 11.
−−
­ä]ÊÓ>®
−− −−
A plane
is flying
Independent
midpoint
of AC
. AB $parallel
BC
12. Aviation
p. Practice
275
For
See
!!". When the
to the !AXB
ground/isalong
AC
Prove:
isosceles.
Exercises Example
TEKS
<
18. y
­Ó£ÞÊʣήÂ
12/7/05 1:59:55 PM
TEKS 12 TAKS1
{ä¨
in tower T measures
angle to
PRACTICE
AND the
PROBLEM
SOLVING
/
2
the
plane
as
40°.
After
the
plane
has
Application
Practice
p.
S31
12.
Aviation
A
plane
is
flying
parallel
Independent
Practice
17–20
3
Find each angle measure.
traveled 2.4 mi to B, the angle to the
For
See
21
4
!!". When the
to the ground
along AC
Exercises Example 13. m∠E
,
14.
m∠TRU
Ó°{ʓˆ
plane is 80°. How can you find BT?
nä¨
plane is at A, an air-traffic controller
12
1
™È
Â
TEKS
TAKS
in tower T measures the angle to
{ä¨
13–16
2
the
plane
as
40°.
After
the
plane
has
/
Skills
Practice
p.
S11
17–20
3
angle to the
xÇÂ
traveled
2.4
mi
to
B,
the
Application Practice p. S31
1
each angle measure.
21
4 Find
/
plane is 80°. How can you find BT?
nä¨
,
13. m∠E
14. m∠TRU
TEKS
TAKS
{ä¨ Â
Ó
­ÈÞÊÊ£®
15. m∠F
16. m∠A
ÊÊÝÊ Ê
Ê ™ÈÂ
­ÎÝÊÊ£ä®Â
/
Skills Practice p. S11
Skills Practice
p. S11
13–16
Application Practice p. S31
_0279.indd 277
0273_0279.indd 277
0273_0279.indd 277
Find each angle measure.
13. m∠E
Find each value. ™ÈÂ
Ó
m∠F
17. 15.
ÚÚâ ÊÊÝÊ Ê
ÊÂ
z
ÊÊ ÊÊÊÊ
Ê£{
ÊÂ
Ó
m∠F
Ó
15.
ÊÊÝÊ Ê
ÊÂ
Find
each value.
-
­ÎÝÊÊ£ä®Â
xÇÂ
14. m∠TRU
­ÎÝÊÊ£ä®Â
16. m∠A
18. y
-
1­Ó£ÞÊʣήÂ
­ÈÞÊÊ£®Â
xÇÂ
­£°xÞÊÊ£Ó®Â1
/
16. m∠A
/
,
­Ó£ÞÊʣήÂ
­ÈÞÊÊ£®Â
ÚÚâ
17. z
18. y
ÊÊ ÊÊÊÊÊ£{
ÊÂ
ÚÚÊÎÊÊÊÊÝÊÊÓ
Ó
8
19. BC
20.
XZ
­Ó£ÞÊʣήÂ
Ó paragraph proof
41. Rewrite
of the
ÓÝ ­£°xÞÊÊ£Ó®
the
Â
Hypotenuse-Leg (HL) Congruence
ÚÚÊxÊÊÊÝÊ ÊÊx
ÚÚÊxÊÊÊÝÊ ÊÊÈ
9
Find
each value.
Ó
Theorem
as
a
two-column
proof.
{
z !ABC
ÓÝ
41. 17.
Rewrite
the paragraph
proof
the triangles. 18. y
Given:
and
!DEF
areof
right
ÊâÊÊÊÊÊ£{
ÊÂ
ÊÚÚ
Ó
<
Hypotenuse-Leg
(HL)
∠C and ∠F
are Congruence
right angles.
­£°xÞÊʣӮ ΠÚÚ
−−
−−
−−
−−
8
19.
BC ACas$aDF
Ê two-column
ÊÊÊÝÊ , and
ÊÓ
Theorem
AB $Pproof.
DE
. midpoint 20. XZ
Þ ÓÝ
­Ó>]ÊÓL® 21. Given:
$ABC
is the
Óis isosceles.
−−
−−
Prove: of
!ABC
$
!DEF
Given:
!ABC
and
!DEF
are
right
triangles.
AB. Q isthe midpoint of AC.
ÚÚÊxÊÊÊÝÊÊx
ÚÚÊxÊÊÊÝÊright
−− are
−− and
−− −− 9
ÊÈ
Ó
∠C
angles.
&
AC∠Fdraw
{
Proof: AB
On
!DEF
EF
.
Mark
G
so
that
FG
=
CB.
Thus
FG
. From the diagram,
%%&
−−
−−
* $ CB+
−−
−−
−− AC−−
−− −−
$ QB
DF
AB $∠F
DE
. right angles. DF
Prove:
&
ÓÝ
⊥ EG by definition
AC PC
$ DF
and, and
∠C and
are
of perpendicular
Ý SAS.
8
< $ !DGF by
Prove:
!DEF
19.
BC !ABC
20.
XZ
ÊÎ$
ÊÊÊ∠DFG
ÝÊ
ÊÓ is a right angle, and ∠DFG
lines.
ThusÚÚ
$
∠C.
!ABC
−−− −−
−−− −− ­ä]Êä®−− ÓÝ
Ó
−− −−
−− ­{>]Êä®
DG On
$ AB
by CPCTC.EF
AB. Mark
$ DE Gasso
given.
DG=
$CB.
DE Thus
by theFG
Transitive
Property.
Proof:
!DEF
that FG
$ CB
%%&
21. −−
Given:−−
$ABC draw
is isosceles.
P is the midpoint
­Ó>]ÊÓL®the diagram,
Þ .From
x −−ÚÚ
−−∠DFG
Ê
Ê
ÊÊ
Ý
Ê
Êx
By
the
Isosceles
Triangle
Theorem
∠G
$
∠E.
$
∠DFE
since
angles
−−
x −−
ÚÚ
9right
ÊÊÊÝand
Ê ÊÈ ∠F
AC $ each
DF
are right
DFor
⊥never
EGÓbytrue.
definition of
perpendicular
ofand
AB
. ∠C
Q Êis
midpoint
ofangles.
AC
.
{ the is
Tell whether
statement
sometimes,
always,
are
congruent.
So
!DGF
$
!DEF
by
AAS.
Therefore
!ABC
$
!DEF
by
the
−−
−−
lines.
∠DFG
is aa sketch.
right angle, and ∠DFG $ ∠C. !ABC $ ÓÝ
!DGF by SAS.
AB &
AC
Support
yourThus
answer
with
−−−
−−−
−−
−−
−−
−−
Transitive
Property.
+
*
−−
−−
DG
$ ABPC
by&CPCTC.
AB $ DE as given. DG $ DE by the <Transitive Property.
Prove:
QB
22. AnBy
equilateral
triangle
is anTheorem
isosceles ∠G
triangle.
the Isosceles
Triangle
$ ∠E. ∠DFG $ ∠DFE since right angles
Ý
Given:
$ABC
is
isosceles.
istriangle
the by
midpoint
­Ó>]ÊÓL®
Þ !DEF
are
congruent.
Soan
!DGF
$P!DEF
Therefore
$
by the
­{>]Êä®
­ä]Êä®
23. 21.
The
vertex
angle
isosceles
is congruent
to !ABC
the
base
angles.
−−AAS.
−− of
of Property.
AB. Q is the midpoint of AC.
Transitive
−− is a right triangle.
−−triangle
24.
An
isosceles
AB
& AC
42.Tell
Lorena
is designing
a window
so that ∠R, always,
∠S, ∠T, or
and
whether
each
statement
is sometimes,
never true.
Óä*Â
+
−−
−−
−− −−
Prove:
PCanswer
&triangle
QB with
∠Uequilateral
are your
right
angles,
VU
$aVT
,obtuse
and m∠UVT
= are
20°.congruent.
sketch.
/
25. Support
An
and
an
triangle
Ý
What
is
m∠RUV?
An equilateral
triangle
an isosceles
triangle.
6
26. 22.
Critical
Thinking
Can aisbase
angle of an
isosceles triangle
angle?
­{>]Êä®
­ä]Êä®be an obtuse
10°
20°
42. Why
Lorena
is
designing
a
window
so
that
∠R,
∠S,
∠T,
and
1
,
Óä
or why
not?
Â
23. The
vertex
angle of
an
isosceles
triangle
is
congruent
to
the
base
angles.
−− −−
∠U whether
are
righteach
angles,
VU $ VTis, sometimes,
and m∠UVT
= 20°. or never
- true.
70°
80° always,
/
Tell
statement
24.
Anisisosceles
triangle
isaa sketch.
right triangle.
What
m∠RUV?
6
Support
your
answer
with
4- 8 Isosceles and Equilateral
Triangles
277
43. Which
of these values of y makes !ABC
isosceles?
20° triangle.
25. An10°
and
obtuse
triangle are congruent.
22.
equilateral
triangle is
anan
isosceles
1
,
1
1
_
_
1
7
{Þ
70°
80°
ÎÞÊÊx
2 ofisancongruent
26.
Can
a base triangle
angle
isosceles triangle
be
an
obtuse angle?
23. Critical
The4vertexThinking
angle of an
isosceles
to the base
angles.
1
1
_
_
Why
or
why
not?
2of these values of y makes !ABC
15 isosceles?
43. 24.
Which
ÞÊÊ£ä An isosceles
triangle is a right triangle.
2
2
1
1
1_
7_
{Þ
An 4equilateral
triangle
an obtuse
triangle
are congruent. ÎÞÊÊx
2 of
44.25.
Gridded
Response
Theand
vertex
angle
an isosceles
1:59:59 PM
4- 8 Isosceles and Equilateral Triangles12/7/05277
1 measures (6t - 9)°, and one
1 the base angles
_
_
triangle
of
2
15
26. Critical Thinking Can a base angle of an isosceles
be an obtuse angle?
triangle
ÞÊÊ£ä
2
2
)°. not?
measures
Find t.
Why or(4t
why
44. Gridded Response The vertex angle of an isosceles
triangle measures (6t - 9)°, and one of the base angles
CHALLENGE
AND EXTEND
4- 8 Isosceles and Equilateral Triangles
measures (4t)°. Find t.
−− −−
−−− −−
45. In the figure, JK $ JL, and KM $ KL. Let m∠J = x°.
ÝÂ
Prove m∠MKL must also be x°.
CHALLENGE AND EXTEND
46. An equilateral !ABC is placed on a coordinate plane.
−− −−
−−− −−
45. In
theside
figure,
JK $measures
JL, and KM
= x°.and
Each
length
2a.$BKL
is .atLet
them∠J
origin,
Prove
mustthe
also
be x°.
coordinates
of A.
C is atm∠MKL
(2a, 0). Find
ÝÂ
277
12/7/05 1:59:59 PM
12/7/05 1:59:59 PM