The second and third tests for similar triangles
Find similar triangles by two sides and the angle between them, or by three sides, work out an unknown side, and avoid the trap of an angle that is not between the sides.
- Ukraine: grade 8
Guide
In the Ukrainian programme this is Grade 8 geometry, §2 "Similarity of triangles", item 14 "The second and third tests for similarity of triangles". The first test (two angles) is already known.
Second test: if two sides of one triangle are proportional to two sides of another and the angles between them are equal, the triangles are similar. Third test: if three sides of one triangle are proportional to three sides of another, the triangles are similar.
Similarity is written △ABC ∾ △A₁B₁C₁ with the vertices in matching order: AB matches A₁B₁, BC matches B₁C₁, AC matches A₁C₁. The ratio of corresponding sides is the coefficient of similarity k.
Important: the equal angle must lie between the proportional sides.