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How are similar triangles used in Euclidean geometry?

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Three theorems of congruent triangles — Euclidean geometry This diagram depicts three basic theorems that triangles are congruent—that is, they are of equal shape and size—if: (1) two sides and the included angle are equal, designated SAS; (2) two angles and the included side are equal, designated ASA; or (3) all three sides are equal, designated SSS.

In Euclidean geometry, similar triangles are figures that share the same shape but may differ in size, with their corresponding sides being proportional and their corresponding angles equal.12

Similar triangles are essential for determining measurements of objects that are difficult to measure directly. By establishing a relationship between a known object and an unknown one—such as using the shadows of a tree and a shorter object of known height—one can use the ratio of their dimensions to calculate the unknown height.1

Key concepts regarding similarity include:

Proportionality: Two triangles are similar if their corresponding sides are in the same proportion, related by a scaling factor s.2
AAA Theorem: The angle-angle-angle (AAA) similarity theorem states that two triangles are similar if and only if their corresponding angles are equal.2
Fundamental Theorem of Similarity: A line segment parallel to one side of a triangle divides the other two sides into proportional segments.2

Beyond practical measurements, similarity is fundamental to trigonometry and is widely applied in construction plans and geographic mapping.2