Similar Triangles And Other Polygons

Similar polygons are polygons whose corresponding angles are equal, and the ratio of the corresponding sides are also equal.
A quadrilateral $ABCD$ is similar to another quadrilateral $PQRS$ if the angles taken in order are equal,
like $\angle A = \angle P,\ \angle B = \angle Q,\ \angle C = \angle R,\ \angle D = \angle S$, and the corresponding sides are in proportions, that is:

$\dfrac{AB}{PQ} = \dfrac{BC}{QR} = \dfrac{CD}{RS} = \dfrac{DA}{SP}$

then, these two quadrilaterals will be similar in shape, although they may be different in size.
In geometry the symbol $\unicode{0x201C}\sim\unicode{0x201D}$ is used to denote similar figures.
In our case we can write:
$ABCD \sim PQRS$

Also, the areas of these figures are in proportion to the ratio of the squares of any two corresponding dimensions, that is:

$\dfrac{Area\ ABCD}{Area\ PQRS} = \dfrac{AB^2}{PQ^2} = \dfrac{BC^2}{QR^2} = \dfrac{CD^2}{RS^2} = \dfrac{DA^2}{SP^2}$ 


In triangles, if corresponding angles are equal, then necessarily their sides are also in proportion. The reverse is also true, that is, if the sides of two triangles are in proportion, then the corresponding angles are equal.

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If two given triangles $ABC$ and $PQR$ are similar with $\angle A = \angle P$, $\angle B = \angle Q$ and $\angle C = \angle R$ then the corresponding sides are in proportion, that is,
$\dfrac{BC}{QR} = \dfrac{AC}{PR} = \dfrac{AB}{PQ}$

$\underline{Rules\ Of\ Triangle\ Similiarity}$
How to decide if two triangles are similar. There is a similarity rule for each of the congruence rules described .

$\underline{SSS}$ If all three sides of two triangles are in proportion then the triangles are similar.

$\underline{SAS}$. If two corresponding sides are in proportion and their included angles are equal then the two triangles are in proportion.

$\underline{AAA}$ If any two angles are equal, then it is easy to see that the third angles are also equal. Hence the triangles are in proportion.

$\underline{RHS}$ If the hypotenuse and any other corresponding side of two right angled triangle are in proportion then the two triangles are similar.

$\underline{Properties\ Of\ Similar\ Polygons}$
If two polygons are similar, then all their corresponding dimensions are in proportion.
If we take the example of two similar triangles with sides $a,\ b,\ c$ and $p,\ q,\ r$, with altitudes as $h_1,\ h_2,\ h_3$ for the first triangle and $H_1,\ H_2,\ H_3$ for the second triangle then:
$\dfrac{h_1}{H_1} = \dfrac{h_2}{H_2} = \dfrac{h_3}{H_3} = \dfrac{a}{p} = \dfrac{b}{q} = \dfrac{c}{r}$

Similarly, the medians, angle bisectors, inradius, circumradius will also be in proportion, and their ratios will be same as the ratio of the sides.