Similar triangles
Similar triangles are triangles whose corresponding angles are congruent. In mathematics, the symbol "~" is used to convey that two or more objects are similar.
![](/img/a/geometry/triangles/similar-triangles/similar-triangles.png)
Similarity theorems
There are a number of different ways to find out if two triangles are similar. The following are a few of the most common.
Angle-Angle (AA) theorem
If two angles in one triangle are congruent to two angles of another triangle, the triangles are similar.
![](/img/a/geometry/triangles/similar-triangles/angle-angle-theorem.png)
In the figure above, if ∠A≅∠D , ∠B≅∠E then, △ABC~△DEF.
Example:
Show that △EFI~△EGH given that
// .![](/img/a/geometry/triangles/similar-triangles/side-angle-side-theorem.png)
Since
// , ∠EFI≅∠G and ∠EIF≅∠H. Therefore, △EFI~△EGH by the AA similarity theorem.Side-Side-Side (SSS) theorem
Two triangles are similar if the lengths of their corresponding sides are proportional.
![](/img/a/geometry/triangles/similar-triangles/side-side-side-theorem.png)
In the figure above, if , then △ABC~△DEF. Note that the corresponding sides do not have to be equal in length.
Side-Angle-Side (SAS) theorem
Two triangles are similar if one of their angles is congruent and the corresponding sides of the congruent angle are proportional in length.
![](/img/a/geometry/triangles/similar-triangles/side-angle-side-theorem.png)
In the figure above, if , and △IEF and △HEG share the same angle, ∠E, then, △IEF~△HEG.