Refresher - Similarity Between Polygons

Properties of Similar Polygons

1. Corresponding angles are the same
   So in the figure above, ${\displaystyle \angle P=\angle L}$, ${\displaystyle \angle Q=\angle M}$, ${\displaystyle \angle R=\angle N}$ and ${\displaystyle \angle S=\angle O}$ .
   From this, it follows that the corresponding exterior angles will also be the same.
2. Corresponding sides are all in the same proportion
   By definition each pair of corresponding sides are in the same proportion, or ratio. So, for example, if in two similar polygons one side is twice the length of the corresponding side in the other, Then all the other sides will be twice the length of their corresponding side also.
3. Corresponding diagonals are in the same proportion
   In each polygon the corresponding diagonals are in the same proportion. Their ratio is the same as the ratio of the sides.
4. Area ratio
   The ratio of the areas of the two polygons is the square of the ratio of the sides. So if the sides are in the ratio 3:1 then the areas will be in the ratio 9:1. This is true for all similar polygons, not just triangles.

Rotation and reflection

Polygons can still be similar even if one of them is rotated, and/or mirror image of the other. In the figure below, all three polygons are similar.

Starting with the original polygon on the left, the center polygon is rotated clockwise 90°, the right one is flipped vertically. Mark the angles in the two right polygons that correspond with the angle P on the left. This is illustrated in more depth for triangles in "Similar Triangles", but is true for all similar polygons, not just triangles.