Chapter 9

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Proportions and Similarity
Using Ratios and
Proportions
A ratio is a comparison of
two numbers by division
An equation that shows
two equivalent ratios
The cross products in a
proportion are equivalent
In the proportion 20 = 2
30 3
20 and 3 are the extremes
and 30 and 2 are the means.
If a = c, then ad = bc.
b d
Similar Polygons
Two polygons are similar if
and only if their
corresponding angles are
congruent and the measures
of their corresponding sides
are proportional.
Used to represent
something that is too large
or too small to be drawn at
actual size.
The ratio of the lengths of
two corresponding sides of
two similar polygons
Similar Triangles
If two angles of one
triangle are congruent to
two corresponding angles
of another triangle, then
the triangles are similar.
If the measures of the sides
of a triangle are proportional
to the measures of the
corresponding sides of
another triangle, then the
triangles are similar.
If the measures of two sides of a
triangle are proportional to the
measures of two corresponding
sides of another triangle and
their included angles are
congruent, then the triangles
are similar.
Proportional Parts and Triangles
If a line is parallel to one
side of a triangle and
intersects the other two
sides, then the triangle
formed is similar to the
original triangle.
If a line is parallel to one
side of a triangle and
intersects the other two
sides, then it separates the
sides into segments of
proportional lengths.
Triangles and Parallel Lines
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.
If a segment joins the
midpoints of two sides of a
triangle, then it is parallel
to the third side, and its
measure equals one-half
the measure of the third
side.
Proportional Parts and
Parallel Lines
If three or more parallel
lines intersect two
transversals, they divide
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.
Perimeters and Similarity
If two triangles are similar,
then the measures of the
corresponding perimeters
are proportional to the
measures of the
corresponding sides.

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