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Similarity
Similar Polygons

Two polygons are similar if these two facts both must be true:

• Corresponding angles are equal.
• The ratios of pairs of corresponding sides must be equal.
(in other words, if they are proportional).
The symbol for "is similar to" is $\sim$

quadrilateral ABCD $\sim$ quadrilateral EFGH

This means:

m<A = m<E, m<B=m<F, m<C=m<G, m<D=m<H and

$\frac{AB}{EF}=\frac{BC}{GH}=\frac{CD}{GH}=\frac{AD}{EH}$

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:

Quadrilaterals that are not similar to one another.

Even though the ratios of corresponding sides are equal, corresponding angles are not equal $(90^o\;\neq\;120^o,\;90^o\;\neq\;60^o)$

Quadrilaterals that are not similar to one another.

Even though corresponding angles are equal, the ratios of each pair of corresponding sides are not equal $(\frac{3}{3}\;\neq\;\frac{5}{3})$

Typically, problems with similar polygons ask for missing sides. To solve for a missing length, find two corresponding sides whose lengths are known. After we do this, we set the ratio equal to the ratio of the missing length and its corresponding side and solve for the variable.

Given that polygon WXYZ $\sim$ polygon ABCD, find the missing measure:

The missing measure m is the lenght of $\bar{XY}$. Write a proportion.

 $\frac{m}{12}=\frac{15}{10}$ XY=m, BC=12, YZ=15, and CD=10 m·10=12·15 Find the cross products. 10m=180 Multiply m=18 Divide each side by 10

Find the scale factor from polygon WXYZ to polygon ABCD

 scale factor: $\frac{YZ}{CD}=\frac{15}{10}\;or\;\frac{3}{2}$ The scale factor is the constant of proportionality

A length on polygon WXYZ is $\frac{3}{2}$ times as long as a corresponding length on polygon ABCD.

Let m represent the measure of $\bar{XY}$:

$m=\frac{3}{2}\;\cdot\;12$

m=18

Congruent Versus Similar polygons
Congruent polygons have the same shape and the same size, while similar figures have the same shape buy may have different sizes.