Similarity and Geometric Mean

Triangles are similar if their corresponding angles are equal and their corresponding sides are in proportion.

AA~ stands for Angle-Angle Similarity, which states that if two angles of one triangle are equal to two angles of another triangle, the triangles are similar.

SSS~ stands for Side-Side-Side Similarity, which states that if the corresponding sides of two triangles are in proportion, the triangles are similar.

SAS~ stands for Side-Angle-Side Similarity, which states that if two sides of one triangle are in proportion to two sides of another triangle and the included angles are equal, the triangles are similar.

The geometric mean of two numbers a and b is given by $$\text{Geometric Mean} = \frac{\text{sqrt}(a \times b)}{1}$$ .

Use the proportion of the corresponding sides: $$\frac{a}{b} = \frac{c}{d}$$ , where a and b are sides of the first triangle and c and d are sides of the second triangle.

The Triangle Proportionality Theorem states that if a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally.

The converse of the Triangle Proportionality Theorem states that if a line divides two sides of a triangle proportionally, then it is parallel to the third side.

To determine if two triangles are similar using SSS~, check if the ratios of the lengths of corresponding sides are equal.

The heights of similar triangles are in the same ratio as the lengths of their corresponding sides.