Two polygons with the same shape are called
similar polygons. The symbol for “is similar to” is ∼. Notice that it is a portion of the “is congruent to” symbol, ≅. When two polygons are similar, these two facts
both must be true:
In Figure
1 , quadrilateral
ABCD ∼ quadrilateral
EFGH.
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Figure 1
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Similar quadrilaterals.
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This means:
m ∠
A =
m ∠
E,
m ∠
B =
m ∠
F,
m ∠
C =
m ∠
G,
m ∠
D =
m ∠
H, and
It is possible for a polygon to have one of the above facts true without having the other fact true. The following two examples show how that is possible.
In Figure
2 , quadrilateral QRST is not similar to quadrilateral WXYZ.
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Figure 2
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Quadrilaterals that are not similar to one another.
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Even though the ratios of corresponding sides are equal, corresponding angles are not equal (90° ≠ 120°, 90° ≠ 60°).
In Figure
3 , quadrilateral
FGHI is not similar to quadrilateral
JKLM.
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Figure 3
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Quadrilaterals that are not similar to one another.
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Even though corresponding angles are equal, the ratios of each pair of corresponding sides are not equal (3/3≠5/3).
Example 1: In Figure
4 , quadrilateral
ABCD ∼ quadrilateral
EFGH. (a) Find
m ∠
E. (b) Find
x.
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Figure 4
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Similar quadrilaterals.
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(a)
m ∠
E = 90° (∠
E and ∠
A are corresponding angles of similar polygons, and corresponding angles of similar polygons are equal.)
(b) 9/6 = 12/
x (If two polygons are similar, then the ratios of each pair of corresponding sides are equal.)
Fundamental Ideas
Similarity
