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Two triangles are
similar if and
only if the corresponding sides are in proportion and the
corresponding angles are congruent. |
There are three accepted methods of proving triangles similar:
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AA |
To show
two triangles are
similar, it is sufficient to show that two angles of one
triangle are congruent (equal) to two angles of the other
triangle. |
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Theorem: |
If two angles
of one triangle are congruent to two angles of another triangle,
the triangles are similar. |
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SSS
for
similarity |
BE
CAREFUL!! SSS for similar triangles is NOT the same
theorem as we used for congruent triangles. To show triangles are
similar, it is sufficient to show that the three sets of
corresponding sides are in proportion. |
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Theorem: |
If the three
sets of corresponding sides of two triangles are in proportion,
the triangles are similar. |
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SAS
for
similarity |
BE
CAREFUL!! SAS for similar triangles is NOT the same
theorem as we used for congruent triangles. To show triangles are
similar, it is sufficient to show that two sets of
corresponding sides are in proportion and the angles they
include are congruent. |
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Theorem: |
If an angle of
one triangle is congruent to the corresponding angle of another
triangle and the lengths of the sides including these angles are
in proportion, the triangles are similar. |
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Once the triangles are similar: |
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Theorem: |
The
corresponding sides of similar triangles are in proportion. |
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Dealing with overlapping triangles: |
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Many
problems involving similar triangles have one triangle
ON
TOP OF
(overlapping)
another triangle.
Since
 is marked to be parallel
to  , we know that we have
<BDE congruent to <DAC
(by corresponding angles). <B is shared by both
triangles, so the two triangles are similar by AA. |
There is an additional theorem that can be
used when working with overlapping triangles: |
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Additional
Theorem: |
If a line is
parallel to one side of a triangle and intersects the other two
sides of the triangle, the line divides these two sides
proportionally. |
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