Similarity of Triangles

Similar triangles have the same shape but not necessarily the same size. The test leans on one fact: if two angles of one triangle equal two angles of another, the third angles must match too, since all three angles in any triangle add to 180 degrees. Once all three angles agree, the triangles are forced into one consistent scale factor. Every side grows or shrinks by that same number, so the sides don't just resemble each other, they lock into one exact ratio.

To use this, identify which side faces which angle, because corresponding sides are the ones across from equal angles. Line those pairs up, write them as a proportion, and solve. A triangle with sides 6 and 9 similar to one with matching sides 10 and x gives 6/10 = 9/x, so x = 15. The same logic handles a line drawn parallel to a triangle's base: that line creates a smaller triangle similar to the original, so the segments it cuts on the two sides are proportional. Sides split into 4 and 6 on one leg and 6 and x on the other give 4/6 = 6/x, so x = 9.

This topic sits next to right-triangle trig and area problems because similarity is the reason trig ratios stay constant for a given angle regardless of triangle size. It also explains real measuring tricks, like using a shadow and a known height to find a building's height, since the sun's rays create two similar triangles at the same instant. Spotting similarity fast often shortens what looks like a hard geometry problem into one proportion.