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How are similar triangles different from congruent triangles?

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Map of the Greco-Roman world highlighting the birthplaces of notable ancient mathematicians, marked with the pi symbol and labeled with their names. Cities such as Athens, Alexandria, Syracuse, and Miletus are shown around the Mediterranean, with modern country boundaries outlined. An inset provides detail for the region of Greece and western Anatolia. — geometry This map spans a millennium of prominent Greco-Roman mathematicians, from Thales of Miletus (c. 600 bce) to Hypatia of Alexandria (c. 400 ce).

In Euclidean geometry, the primary distinction between similar and congruent triangles lies in their size and the relationship between their corresponding parts.12

Congruent Triangles

Two triangles are congruent if they have the exact same shape and size. They can be perfectly superimposed on one another through rigid motion. To prove congruence, one must demonstrate that corresponding parts are equal in measure, typically using established theorems such as:123
* Side-Angle-Side (SAS)3
* Angle-Side-Angle (ASA)3
* Side-Side-Side (SSS)3

Similar Triangles

Similar triangles share the same shape but may differ in size. While their corresponding angles are equal, their corresponding sides are not necessarily equal in length; instead, they are proportional. Two triangles are similar if they are related by a scaling factor s, meaning if one triangle has sides a, b, and c, the other has sides sa, sb, and sc. This relationship is fundamental to trigonometry and allows for the measurement of objects that are otherwise difficult to assess directly.12