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7.4 Skill Builder
Sum of the Angles in a Triangle
In any triangle, the sum of the angle measures is 180°.
So, to find an unknown angle measure:
• start with 180°
• subtract the known measures
An isosceles triangle has 2 equal sides and
To find the measure of the third angle,
subtract the measure of the
equal angles twice.
2 equal angles.
To find the measure of each equal angle,
subtract the known angle from 180°,
then divide by 2.
G
A
40°
C
50°
B
!A ! 180° " 50° " 50°
! 80°
J
H
Sum of equal angles is: 180° " 40° ! 140°
Measure of each equal angle: 140° # 2 ! 70°
Check
1. Find the measure of the third angle.
a) D
b)
P
60°
Q
E
65°
50°
R
F
50 " _____
60
!E ! 180° " _____
70
! _____
180-65-65
!Q ! ___________________
50
! _____
2. Find the measure of each equal angle.
S
Sum of equal angles is:
110
70 ! _____
180° " _____
70°
Measure of each equal angle:
T
110 # 2 ! 55
_____
_____
U
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7.4 Similar Triangles
FOCUS Use the properties of similar triangles to solve problems.
A triangle is a special polygon.
When two triangles are similar:
• matching angles are equal OR
• matching sides are proportional
The order in which similar triangles are named gives a lot of information.
Suppose !ABC ~ !DEF.
The symbol ~ means
“is similar to.”
Then, "A ! "D, "B ! "E, and "C ! "F
Similarly, AB matches DE, BC matches EF, and AC matches DF.
Example 1
Identifying Similar Triangles
Name the similar triangles.
4.8 cm
D
3.0 cm
E
3.6 cm
F
Solution
Y
2.0 cm
X
2.4 cm
3.2 cm
Z
Angle measures are not given.
So, find out if matching sides are proportional.
In !DEF, order the sides from shortest to longest: FD , EF , DE
In !XYZ, order the sides from shortest to longest: XY , YZ , ZX
Find the scale factors of matching sides.
length of FD
length of XY
! 3.0 cm
2.0 cm
! 1.5
length of EF
length of YZ
! 3.6 cm
2.4 cm
! 1.5
length of DE
length of ZX
! 4.8 cm
3.2 cm
! 1.5
Since all scale factors are the same, the triangles are similar.
The longest and shortest sides meet at vertices: D and X
The two longer sides meet at vertices:
E and Z
The two shorter sides meet at vertices:
F and Y
So, !DEF ~ !XZY
Read the letters down
the columns.
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Check
1. In each diagram, name two similar triangles.
a) Two angles in each triangle are given.
The measure of the third angle
in each triangle is:
36 - 68 = 76
180° ! _________________
A
R
B
68°
36°
36°
68°
C
Q
P
List matching angles:
!A " _____
<P " _____
76
<Q " _____
!B " _____
68
!C " _____
36
<R " _____
ARE equal.
Matching angles _______
ARE similar.
So, the triangles _______
To name the triangles, order the letters so matching angles correspond.
PQR
"ABC ~ "_______
b) Find out if matching sides are proportional.
In "DEF, order the sides from shortest to longest:
D
J
5.4 cm
EF, DE, FD
_______________
In "JKL, order the sides from shortest to longest:
JK, KL, LJ
_______________
Find the scale factors of matching sides.
length of EF
length of JK
CM
" 2.8
1.4 CM
length of DE
length of KL
"
length of
length of
FD
"
LJ
3.2 cm
2.7 cm
K
1.6 cm
L
E
2
" ____
3.2 CM
2
" ____
1.6 CM
5.4 CM
2
2.7 CM " ____
THE SAME. So, the triangles _______________
ARE SIMILAR .
All scale factors are ___________
D and ____
L
The two longer sides meet at vertices:
____
K
The two shorter sides meet at vertices:
E and ____
____
K
F and ____
The longest and shortest sides meet at vertices:
____
So, "DEF ~ "______
LKJ
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1.4 cm
2.8 cm
F
MMS9_Prep book_Unit7.4.qxp
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Using Similar Triangles to Determine a Length
These two triangles are similar.
Find the length of TU.
S
°
6 cm
Q
3 cm
°
T
U
R
2 cm
P
Solution
List matching angles:
!S ! !P
!T ! !Q
So, "STU ~ "PQR
!U ! !R
"STU is an enlargement of "PQR.
Choose a pair of matching sides
whose lengths are both known:
SU ! 6 cm and PR ! 2 cm
Scale factor ! length on enlargement
length on original
Consider the triangle
with the unknown
length as a reduction
or enlargement of the
other triangle.
! 6 cm
2 cm
!3
The scale factor is 3.
Use the scale factor to find the length of TU.
TU and QR are matching sides.
Length of QR: 3 cm
Scale factor: 3
Length of TU: 3 " 3 cm ! 9 cm
So, TU has length 9 cm.
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Check
1. These two triangles are similar.
Find the length of XV.
V
F
20 cm
X °
W
2 cm
10 cm
H
G
°
List matching angles:
<X
<V
!F ! _____
!G ! _____
XVW
So, "FGH ~ "_______
<W
!H ! _____
XVW is a reduction of _________
FGH
.
_________
Choose a pair of matching sides whose lengths are both known:
FH = 10 CM AND XW = 2 CM
__________________________________
Scale factor ! length on reduction
length on original
! 2 CM
10 CM
! _____
0.2
0.2 .
The scale factor is _____
Use the scale factor to find the length of XV.
XV and FG are matching sides.
20 CM
Length of FG: ________
0.2
Scale factor: ______
0.2 X 20 CM = 4 CM
Length of XV: _________________________
4 CM .
So, XV has length ______
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Practice
T
1. In each diagram, name two similar triangles.
40°
F
a) Two angles in each triangle are given.
The measure of the third angle
110 - 40 = 30
in each triangle is: 180° ! ___________________
40°
G
U
110°
110°
H
S
List matching angles:
110
40
<S " _____
<T " _____
<U
!F " _____
" _____
!G " _____
!H " _____
30
ARE equal, so, the triangles _______
ARE similar.
Matching angles _______
UTS
To name the triangles, order the letters so matching angles correspond. "FGH ~ "______
b) Find out if matching sides are proportional.
JK, KL, LJ
In "JKL, order the sides from shortest to longest: ______________
QR, SQ, RS
In "QRS, order the sides from shortest to longest: ______________
Find the scale factors of matching sides.
length of JK
length of QR
3.3 CM
"
2.2 CM
length of
length of
KL
4.8 CM
SQ " 3.2 CM
length of
length of
LJ
"
RS
K
1.5
" ____
Q
4.8 cm
3.2 cm
L
3.3 cm
S
1.5
" ____
5.7 cm
2.2 cm
3.8 cm
R
J
5.7 CM
1.5
3.8 CM " ____
ARE SIMILAR
EQUAL
All scale factors are ____________
. So, the triangles ____________
.
R
J and ____
The longest and shortest sides meet at vertices: ____
Q
K and ____
The two shorter sides meet at vertices:
____
The two longer sides meet at vertices:
S
L and ____
____
RQS
So, "JKL ~ "______
2. Are these two triangles similar?
P
In "PQR, order the sides from shortest to longest:
QR, RP, PQ
________________
In "BCD, order the sides from shortest to longest:
CD, DB, BC
________________
Find the scale factors of matching sides.
length of QR
length of CD
B
12 cm
8 cm
Q
6 cm
R
4 cm
5 cm
D 2 cm C
6 CM
3
" ____
2 CM
8 CM
length of RP
"
" ____
2
length of DN
4 CM
12 CM
length of PQ
2.4
"
" ____
length of BC
5 CM
"
ARE NOT SIMILAR.
DIFFERENT . So, the triangles __________________
All scale factors are ______________
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3. These two triangles are similar.
Find the length of EC.
G
C
List matching angles:
D
<H
<G
!C ! ______
!D ! ______
GHF
So, "CDE ~ "_______
2 cm
<F
!E ! ______
°
4 cm
E
F
°
H
5 cm
GHF .
CDE
is a reduction of ________
________
Choose a pair of matching sides whose lengths are both known:
DE = 2M AND HF = 5 CM
__________________________________
Scale factor ! length on reduction
length on original
!
2 CM
5 CM
! _____
0.4
0.4 .
The scale factor is _____
Use the scale factor to find the length of EC.
FG are matching sides.
EC and _____
4 CM
FG : _______
Length of _____
0.4
Scale factor: _____
0.4 X 4 CM = 1.6 CM
Length of EC: __________________________
1.6 CM .
So, EC has length ________
4. At a certain time of day, two trees cast shadows.
Find the height of the taller tree.
Y
B
EQUAL .
Matching angles are _________
XYZ
So, "ABC ~ "_______
ENLARGEMENT
"XYZ is an ________________ of "ABC.
ZX = 3.6 M AND CA = 2M
Use sides _____________________________
to find the scale factor.
length on enlargement
length on original
!
5.4 m
3m
55°
A
2m
C
3.6 M
2M
1.8
! _____
The scale factor is 1.8.
Use the scale factor to find the height of the taller tree, YZ.
BC and YZ are matching sides.
1.8
3M
Length of BC: ______
Scale factor: _____
1.8 X 3 M = 5.4 M
Length of YZ: _____________________
5.4M
So, the height of the taller tree is _____________.
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X
55°
3.6 m
Z
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5. The two triangles in this diagram are similar.
Find the length of DE.
A
1.8 cm
E
D °
1.2 cm
To better see the individual triangles,
we draw the triangles separately.
B °
4.0 cm
A
C
A
1.8 cm + 1.2 cm =
——
cm
D °
———
E
cm
B °
——
cm
C
<D
<E
<A
!A ! ______
!B ! ______
!C ! ______
ADE
So, "ABC ~ "______
ABC
ADE
.
_________ is a reduction of _________
Choose a pair of matching sides whose lengths are both known:
AD = 1.8 CM AND AB AB = 3.0 CM
__________________________________
Scale factor ! length on reduction
length on original
1.8 CM
! 3.0 CM
0.6
! _____
0.6 .
The scale factor is _____
Use the scale factor to find the length of DE.
BC are matching sides.
DE and _____
_____
4.0 CM
BC : __________
Length of _____
0.6
Scale factor: _____
0.6 X 4.0 CM = 2.4 CM
Length of DE: __________________________
So, DE has length _________
2.4 CM .
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