Chapter 8 Similarity

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Chapter 8Similarity

• Section 8.5
• Proving Triangles are Similar

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?A ???D and ?C ???F? ?
?ABC ?DEF
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THEOREM 8.2 Side-Side-Side (SSS) Similarity
Theorem
If the corresponding sides of two triangles are
proportional, then the triangles are similar.
then ?ABC ?PQR.
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P
Q
SOLUTION
Paragraph Proof
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.
Use the definition of congruent triangles and the
AA Similarity Postulate to conclude that ? RST
? LMN.
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Determine if the triangles are similar
Compare Side Lengths of ?LKM and ?NOP
Ratios Different, triangles not similar
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Determine if the triangles are similar
Compare Side Lengths of ?LKM and ?NOP
Ratios Same, triangles are similar ?RQS ?LKM
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THEOREM 8.3 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.
then ?XYZ ?MNP.
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?CED
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Similar triangles can be used to find distances
that are difficult to measure directly.
ROCK CLIMBING You are at an indoor climbing
wall. To estimate the height of the wall, you
place a mirror on the floor 85 feet from the base
of the wall. Then 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.
Not drawn to scale
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Use similar triangles to estimate the height of
the wall.
SOLUTION
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.
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Use similar triangles to estimate the height of
the wall.
SOLUTION
Ratios of lengths of corresponding sides are
equal.
So, the height of the wall is about 65 feet.
Substitute.
Multiply each side by 5 and simplify.
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The Tree is 72 feet tall
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72
The Tree is 72 feet tall
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x
The mirror would need to be placed 36 feet from
the tree
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