Similar polygons are polygons with the same shape, but not necessarily the same size (and thus are dilations of each other). We can use proportions and scale factors to finding missing parts of similar polygons. We can use the symbol ~ to denote similarity.
For generic polygons, in order to prove similarity, you have to show that all pairs of corresponding angles are the same, and all pairs of corresponding sides are ratios of each other.
Similar Triangles
With triangles, similarity has shortcuts, as shown in the following postulates:
- AA Similarity Postulate: Two triangles are similar if two angles of one triangle are congruent to two triangles of the second triangle.
- SSS Similarity Postulate: Two triangles are similar if their three corresponding side lengths are proportional.
- SAS Similarity Postulate. Two triangles are similar if two of their corresponding side lengths are proportional with the included angles of the triangles congruent.
Note that if two triangles are congruent, then technically the are also similar.
Here are some examples of similar triangles:
The triangles are similar via SSS~ since $ \displaystyle \frac{{10}}{6}=\frac{{10}}{6}=\frac{{10}}{6}$.
The triangles are similar via SAS~, since $ \displaystyle \frac{{15}}{9}=\frac{{10}}{6}$, and both triangles have an included angles of $ \displaystyle 90{}^\circ $.
Triangle $ ABC$ is similar to triangle $ CBD$ are similar via SAS~, since $ \displaystyle \frac{{10}}{4}=\frac{{25}}{10}$, and triangles have congruent included angles.
Triangle $ ABD$ is similar to triangle $ CBD$ are similar via AA~.
Side-Splitter Theorem:
The Triangle Proportionality Theorem, or Side-Splitter Theorem states that for a line parallel to one side of a triangle that intersects the other two sides, that line divides those two sides proportionally.
This creates a smaller triangle within a larger triangle, and the two triangles are similar.
Here’s an example:
Find $ x$: 
Problems and Solutions:
Problem: