Download 8.4 Similar Triangles

10.2 Proving Triangles Similar
Geometry
Mr. Calise
Objectives/Assignment
Identify similar triangles.
 Use similar triangles in real-life problems
such as using shadows to determine the
height of the Great Pyramid

Identifying Similar Triangles

In this lesson, you will continue the study
similar polygons by looking at the
properties of similar triangles.
Ex. 1: Writing Proportionality
Statements

In the diagram,
∆BTW ~ ∆ETC.
a. Write the statement
of proportionality.
b. Find mTEC.
c. Find ET and BE.
T
E
34°
C
3
20
79°
B
12
W
Ex. 1: Writing Proportionality
Statements

In the diagram,
∆BTW ~ ∆ETC.
a. Write the statement
of proportionality.
ET
BT
=
TC
TW
=
T
E
34°
C
3
20
CE
WB
79°
B
12
W
Ex. 1: Writing Proportionality
Statements

In the diagram,
∆BTW ~ ∆ETC.
Find mTEC.
B  TEC, SO
mTEC = 79°
b.
T
E
34°
C
3
20
79°
B
12
W
Ex. 1: Writing Proportionality
Statements

In the diagram,
∆BTW ~ ∆ETC.
c. Find ET and BE.
CE
ET
=
WB
BT
Write proportion.
3
ET
=
12
20
Substitute values.
3(20)
= ET
12
5
= ET
T
E
C
3
20
Multiply each side by 20.
Simplify.
34°
79°
B
12
W
Because BE = BT – ET, BE = 20 – 5 = 15. So, ET is 5 units and BE is
15 units.
Postulate 25 Angle-Angle Similarity
Postulate

If two angles of one
triangle are
congruent to the two
angles of another
triangle, then the two
triangles are similar.
 If JKL  XYZ and
KJL  YXZ, then
∆JKL ~ ∆XYZ.
K
L
Y
J
X
Z
Ex. 2: Proving that two triangles are
similar

Color variations in
the tourmaline
crystal shown lie
along the sides of
isosceles triangles.
In the triangles, each
vertex measures
52°. Explain why the
triangles are similar.
Ex. 2: Proving that two triangles are
similar

Solution. Because
the triangles are
isosceles, you can
determine that each
base angle is 64°.
Using the AA
Similarity Postulate,
you can conclude the
triangles are similar.
Side-Angle-Side Similarity Theorem
If an angle in one triangle is congruent to
an angle in another triangle, and the
sides including the two angles are
proportional, then the two triangles are
similar.
 (SAS Similarity Thm.)

USING SIMILARITY THEOREMS
THEOREM S
THEOREM 10.1 Side-Angle-Side (SAS) Similarity Theorem
If an angle of one triangle is
congruent to an angle of a
second triangle and the lengths
of the sides including these
angles are proportional, then the
triangles are similar.
If
X
P
XY
M and ZX =
PM
then XYZ ~ MNP.
M
X
MN
Z
Y
N
Side-Side-Side Similarity Theorem
If the corresponding sides of two
triangles are proportional, then the two
triangles are similar.
 (SSS Similarity Thm.)

USING SIMILARITY THEOREMS
THEOREM S
THEOREM 10.2 Side-Side-Side (SSS) Similarity Theorem
If the corresponding sides of two
triangles are proportional, then the
triangles are similar.
P
A
Q
R
If AB = BC = CA
PQ
QR
RP
then ABC ~ PQR.
B
C
Using the SSS Similarity Theorem
Which of the following three triangles are similar?
12
A
6
C
E
6
9
F
14
G
J
4
8
6
10
D
B
H
SOLUTION
To decide which of the triangles are similar, consider the
ratios of the lengths of corresponding sides.
Ratios of Side Lengths of  ABC and  DEF
AB
= 46 = 32 ,
DE
Shortest sides
CA
3
= 12
=
,
8
2
FD
Longest sides
BC
= 69 = 32
EF
Remaining sides
Because all of the ratios are equal,  ABC ~  DEF
Finding Distance Indirectly
ROCK CLIMBING
are at an
indoor
climbing
wall.
To estimate
of
SimilarYou
triangles
can
be used
to find
distances
thatthe
areheight
difficult
the wall, youto
place
a mirror
on the floor 85 feet from the base of the wall. Then
measure
directly.
you walk backward until you can see the top of the wall centered in the mirror.
You are 6.5 feet from the mirror and your eyes are 5 feet above the ground.
Use similar triangles to estimate
the height of the wall.
D
B
5 ft
Not drawn to scale
A
6.5 ft
C
85 ft
E
Finding Distance Indirectly
Use similar triangles to estimate
the height of the wall.
SOLUTION
Due to the reflective property of mirrors,
you can reason that ACB 
ECD.
D
Using the fact that  ABC and  EDC
are right triangles, you can apply the
AA Similarity Postulate to conclude
that these two triangles are similar.
B
5 ft
A
6.5 ft
C
85 ft
E
Finding Distance Indirectly
Use similar triangles to estimate
the height of the wall.
SOLUTION
DE EC
Ratios of lengths of
=
corresponding sides are equal.
BA AC
So,
the height of the wall is about 65 feet.
DE = 85
Substitute.
5
6.5
D
Multiply each side by
5 and simplify.
65.38  DE
B
5 ft
A
6.5 ft
C
85 ft
E
Note:

If two polygons are similar, then the ratio
of any two corresponding lengths (such
as altitudes, medians, angle bisector
segments, and diagonals) is equal to the
scale factor of the similar polygons.
Ex. 5: Using Scale Factors


Find the length of the altitude QS.
Solution: Find the scale factor of
∆NQP to ∆TQR.
NP
=
TR
12+12
8+8
=
QS
3
=
2
Substitute 6 for QM and solve for
QS to show that QS = 4
12
M
24 = 3
16
2
Now, because the ratio of the
lengths of the altitudes is equal to
the scale factor, you can write the
following equation:
QM
12
N
P
6
Q
R
8
S
8
T
Homework
Finish The Worksheets from Thursday
 DUE on Monday!
