G9 Math Q3- Week 8- Proving Triangle Similarity.ppt

similar triangle

1. Proving Triangles are
Similar
GEOMETRY

2. Objectives:
Use similarity theorems to prove that two triangles
are similar
Use similar triangles to solve real-life problems such
as finding the height of a climbing wall.

3. Using Similarity Theorems
In this lesson, you will study 2 alternate
ways of proving that two triangles are
similar: Side-Side-Side Similarity Theorem
and the Side-Angle-Side Similarity
Theorem. The first theorem is proved in
Example 1 and you are asked to prove the
second in Exercise 31.

4. Side Side Side(SSS)
Similarity Theorem
If the corresponding sides of two triangles are
proportional, then the triangles are similar.
A
B
C
P
Q
R
AB
PQQRRP
BCCA
= =
THEN ABC ~ PQR
∆ ∆

5. Side Angle Side Similarity
Thm.
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.
X
Z Y
M
P N
If X  M
and
ZX
PM
=
XY
M
N
THEN XYZ ~ MNP
∆ ∆

6. Ex. 1: Proof of Theorem 8.2
Given: Prove
RS
LMM
N
NL
STTR
= =
∆RST ~ LMN
∆
Locate P on RS so that PS = LM. Draw PQ so that PQ ║ RT.
Then ∆RST ~ ∆PSQ, by the AA Similarity Postulate, and
RS
LMM
N
NL
STTR
= =
Because PS = LM, you can substitute in
the given proportion and find that SQ =
MN and QP = NL. By the SSS Congruence
Theorem, it follows that PSQ
∆
 LMN
∆
Finally, use the definition of congruent
triangles and the AA Similarity Postulate
to conclude that RST
∆
~ LMN.
∆

7. Ex. 2: Using the SSS Similarity Thm.Which of the three triangles are similar?
96
12A
B
C
6
4
8
D
E
F
106
14
G
H
J
To decide which, if any, of the triangles are similar, you
need to consider the ratios of the lengths of
corresponding sides.
Ratios of Side Lengths of ABC and DEF.
∆ ∆
AB
DE4 2
6 3
= =
CA
FD8 2
123
= =
BC
EF6 2
9 3
= =
Because all of the ratios are equal, ABC
∆
~ DEF.
∆

8. Ratios of Side Lengths of ∆ABC ~ ∆GHJ
AB
GH6
1
6
= =
CA
JG147
126
= =
BC
HJ10
9
=
Because the ratios are not equal, ABC
∆
and GHJ are not similar.
∆
Since ABC is similar to DEF and ABC is
∆ ∆ ∆
not similar to GHJ, DEF is not similar to
∆ ∆
GHJ.
∆

9. Ex. 3: Using the SAS Similarity Thm.Use the given lengths to prove that ∆RST ~ ∆PSQ.
15
12
54
S
R T
P Q
Given: SP=4, PR = 12, SQ
= 5, and QT = 15;
Prove: RST
∆
~ PSQ
∆
Use the SAS Similarity
Theorem. Begin by finding
the ratios of the lengths of
the corresponding sides.
SR
SP
SP + PR
SP
4 + 12
4
= = =
16
4
=4

10. ST
SQ
SQ + QT
SQ
5 + 15
5
= = =
20
5
=4
So, the side lengths SR and ST are
proportional to the corresponding side
lengths of PSQ. Because
∆
S is the
included angle in both triangles, use the SAS
Similarity Theorem to conclude that RST
∆
~
PSQ.
∆

11. Using Similar Triangles in Real Life
Ex. 6 – Finding Distance indirectly.
To measure the width of a river, you use a surveying
technique, as shown in the diagram.
9
12
63

12. 9
12
63Solution
By the AA Similarity
Postulate, PQR
∆
~ STR.
∆
RQ
RTST
PQ
=
RQ
129
63
=
RQ12 7
●
=
Write the
proportion.
Substitute.
Solve for TS.RQ84=
Multiply each side by
12.
So the river is 84 feet wide.

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