8.5 Proving Triangles are Similar

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8.5 Proving Triangles are Similar

• Geometry

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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.
• Assignment pp.492-493 1-26

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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.

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Side Side Side(SSS) Similarity Theorem

• If the corresponding sides of two triangles are
  proportional, then the triangles are similar.

THEN ?ABC ?PQR
AB
BC
CA


PQ
QR
RP
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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.

ZX
XY
If ?X ? ?M and

PM
MN
THEN ?XYZ ?MNP
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Ex. 1 Proof of Theorem 8.2

• Given

• Prove

RS
ST
TR
?RST ?LMN


LM
MN
NL
Locate P on RS so that PS LM. Draw PQ so that
PQ RT. Then ?RST ?PSQ, by the AA Similarity
Postulate, and
RS
ST
TR


LM
MN
NL
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.
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Ex. 2 Using the SSS Similarity Thm.

• Which of the three triangles are similar?

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
6
3
CA
12
3
BC
9
3






DE
4
2
FD
8
2
EF
6
2
?Because all of the ratios are equal, ?ABC ?DEF.
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Ratios of Side Lengths of ?ABC ?GHJ
AB
6
CA
12
6
BC
9
1





GH
6
JG
14
7
HJ
10
?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.
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Ex. 3 Using the SAS Similarity Thm.

• Use the given lengths to prove that ?RST ?PSQ.

Given SP4, 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 PR
4 12
16

4



SP
SP
4
4
10
ST
SQ QT
5 15
20

4



SQ
SQ
5
5
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.
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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.

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Solution
By the AA Similarity Postulate, ?PQR ?STR.
RQ
PQ

Write the proportion.
RT
ST
RQ
63

Substitute.
12
9
RQ
12 ? 7

Multiply each side by 12.
Solve for TS.
RQ
84

?So the river is 84 feet wide.