Proportional Parts

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Proportional Parts
Lesson 7-4
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Two polygons are similar if and only if their
corresponding angles are congruent and the
measures of their corresponding sides are
proportional.
Similar Polygons
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Triangle Proportionality Theorem
If a line is parallel to one side of a triangle
and intersects the other two sides in two
distinct points, then it separates these sides
into segments of proportional length.
Converse
If a line intersects two sides of a triangle and
separates the sides into corresponding segments
of proportional lengths, then the line is
parallel to the third side.
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Examples
If BE 6, EA 4, and BD 9, find DC.
Example 1
6x 36
x 6
Solve for x.
Example 2
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Theorem
A segment that joins the midpoints of two sides
of a triangle is parallel to the third side of
the triangle, and its length is one-half the
length of the third side.
R
M
L
T
S
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Corollary
If three or more parallel lines have two
transversals, they cut off the transversals
proportionally.
If three or more parallel lines cut off congruent
segments on one transversal, then they cut off
congruent segments on every transversal.
F
E
D
A
B
C
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Theorem
An angle bisector in a triangle separates the
opposite side into segments that have the same
ratio as the other two sides.
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If two triangles are similar
(1) then the perimeters are proportional to the
measures of the corresponding sides.
(2) then the measures of the corresponding
altitudes are proportional to the measure of the
corresponding sides..
(3) then the measures of the corresponding angle
bisectors of the triangles are proportional to
the measures of the corresponding sides..
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Example
Given ?ABC ?DEF, AB 15, AC 20, BC 25,
and DF 4. Find the perimeter of ?DEF.
The perimeter of ?ABC is 15 20 25 60. Side
DF corresponds to side AC, so we can set up a
proportion as