Similarity
Lessons in this chapter
About Similarity
Similarity extends the similarity idea from Transformations into a fully provable, testable relationship between two polygons.
Similarity of Polygons defines similar polygons as having proportional corresponding sides and equal corresponding angles, before Triangle Similarity introduces shortcut theorems (like AA) for proving two triangles similar without checking every side and angle.
Proofs using Similarity closes the chapter by using proven similarity to solve for unknown side lengths, or as one step within a larger geometric proof.
This is a three lesson chapter and each lesson opens as a Geometry worksheet in the page: Similarity of Polygons, Triangle Similarity, and Proofs using Similarity. The questions are scored as you answer them, so a proportion set up with corresponding sides in different orders is caught before it becomes a habit. Free to try, with no login needed for your first 3.
Frequently asked questions
Similarity of Polygons defines similar figures as having equal corresponding angles and proportional corresponding sides, and introduces the scale factor between them. Triangle Similarity gives the shortcuts, AA, SSS and SAS, that establish similarity from a small amount of information, and Proofs using Similarity uses those results to prove side relationships, including the ones behind the geometric mean in a right triangle. The chapter sits in the high school Geometry conceptual category (G).
Similar polygons have all corresponding angles equal and all corresponding sides in the same constant ratio (the scale factor): same shape, but not necessarily the same size.
AA (angle-angle) similarity says if two angles of one triangle equal two angles of another, the triangles are similar, since a triangle's three angles always sum to 180°, matching two angles automatically forces the third to match too.
Set up a proportion between corresponding sides of the two similar triangles, using the known sides and the missing one, then solve for the unknown.
Congruent figures are identical in both shape AND size; similar figures share the same shape but can differ in size, related by a scale factor rather than being an exact match.
Proven similarity between two triangles (or other figures) can establish a proportional relationship between their sides, which is often exactly the fact needed to complete a larger proof about lengths or areas.