Similar triangles

Similar triangles have equal corresponding angles and proportional corresponding side lengths. They have the same shape but may differ in size or orientation.

Triangle: A polygon with three sides and three interior angles. Similarity compares the angles and side lengths of these basic polygons.
Angle-angle similarity: A criterion stating that two triangles are similar when two pairs of corresponding angles are equal. The third angle pair must then also be equal.
Indirect measurement: A method of determining a quantity from measurements of related quantities rather than measuring it directly. Similar triangles can determine heights or distances that are difficult to measure directly.
Dilation: A transformation that enlarges or reduces a figure by a fixed factor from a center. A dilation maps a triangle to a similar triangle.
Corresponding angles: Angles occupying matching positions in two geometric figures or across a transversal. Equal corresponding angles establish the angle condition for triangle similarity.
Side-side-side similarity: A criterion stating that triangles are similar when their corresponding side lengths are proportional. It tests similarity directly through all three side ratios.
Triangulation: A method of locating a point by measuring directions or distances from known points. Triangle geometry supports distance estimates when direct measurement is impractical.
Homothety: A transformation that scales points by a common factor from a fixed center. It provides a geometric transformation connecting similarly oriented triangles.
Proportionality: A relation in which quantities have a constant ratio. Corresponding side lengths of similar triangles share one constant ratio.
Side-angle-side similarity: A criterion stating that triangles are similar when two corresponding side pairs are proportional and their included angles are equal. The included angle and two side ratios suffice to establish similarity.

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