Transcript Ch7-Sec7.4

7
THE NATURE OF
GEOMETRY
Copyright © Cengage Learning. All rights reserved.
7.4
Similar Triangles
Copyright © Cengage Learning. All rights reserved.
Similar Triangles
Congruent figures have exactly the same size and shape.
However, it is possible for figures to have exactly the same
shape without necessarily having the same size.
Such figures are called similar figures. If ABC is similar to
DEF, we write
ABC ~ DEF
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Similar Triangles
Similar triangles are shown in Figure 7.38.
Similar Triangles
Figure 7.38
m A = m D, so these are corresponding angles.
m B = m E, so these are corresponding angles.
m C = m F, so these are corresponding angles.
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Similar Triangles
Side BC is opposite A and side EF is opposite
so we say that BC corresponds to EF.
D,
AC corresponds to DF.
AB corresponds to DE.
The corresponding angles of similar triangles are those
angles that have equal measure. The corresponding sides
are those sides that are opposite equal angles.
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Similar Triangles
Even though corresponding angles are equal,
corresponding sides do not need to have the same length.
If they do have the same length, the triangles are
congruent.
However, when they are not the same length, we can say
they are proportional. From Figure 7.38 we see that the
lengths of the sides are labeled a, b, c and d, e, f.
Similar Triangles
Figure 7.38
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Similar Triangles
When we say the sides are proportional, we mean
Primary ratios:
Reciprocals:
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Similar Triangles
We summarize with an important property of similar
triangles called the similar triangle theorem.
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Example 2 – Find lengths given similar triangles
Given the similar triangles in Figure 7.40, find the unknown
lengths marked b and c.
Given ABC ~ ABC
Figure 7.40
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Example 2 – Solution
Since corresponding sides are proportional (other
proportions are possible), we have
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Similar Triangles
There is a relationship between the sizes of the angles of a
right triangle and the ratios of the lengths of the sides.
In a right triangle, the side opposite the right angle is called
the hypotenuse. Each of the acute angles of a right
triangle has one side that is the hypotenuse; the other side
of that angle is called the adjacent side. (See Figure 7.42.)
A right triangle
Figure 7.42
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Similar Triangles
In ABC with right angle at C:
The hypotenuse is c;
The side adjacent to A is b;
The side adjacent to B is a.
The side opposite
A is a, and the side opposite
B is b.
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