Proving triangles similar
For triangles, similarity can be proved with less information than for other polygons. Three shortcut tests make similar triangles a powerful tool for measuring the unmeasurable.
Angle-angle (AA)
If two angles of one triangle are congruent to two angles of another, the triangles are similar. The third angles must match too, since each triangle's angles sum to 180 degrees. AA is the most commonly used similarity test.
Side-side-side (SSS) similarity
If the three sides of one triangle are proportional to the three sides of another, the triangles are similar. Triangles with sides 3, 4, 5 and 9, 12, 15 are similar with scale factor 3.
Side-angle-side (SAS) similarity
If two sides of one triangle are proportional to two sides of another and the included angles are congruent, the triangles are similar. Sides 4 and 6 with a 50-degree angle between them, and sides 6 and 9 with a 50-degree angle between them, give similar triangles with scale factor 1.5.
Similar versus congruent
Congruent triangles are similar with scale factor 1. SSS and SAS similarity require proportional rather than equal sides. AA has no congruence counterpart, since matching angles fix the shape but not the size.
Indirect measurement with shadows
At the same moment, the Sun's rays make the same angle with the ground everywhere nearby, so an object and its shadow form a triangle similar to any other object and its shadow. A 1.8 m person casting a 2.4 m shadow next to a tree casting a 20 m shadow means the tree is 1.8 x 20 / 2.4 = 15 m tall.
Thales and the pyramid
According to ancient accounts, the Greek mathematician Thales of Miletus, around 600 BCE, measured the height of an Egyptian pyramid by comparing its shadow with that of a stick, using exactly this similar-triangle reasoning.
Similar triangles in a right triangle
Drawing the altitude to the hypotenuse of a right triangle creates two smaller triangles similar to the original and to each other, by AA. This leads to the geometric mean relationships and to one of the many proofs of the Pythagorean theorem.
Worked example
To measure a pond's width, a surveyor sets up triangles ABC and ADE with DE parallel to BC. Since DE is parallel to BC, corresponding angles match and the triangles are similar by AA. If AD = 30 m, AB = 90 m, and DE = 12 m, then BC = 12 x 90 / 30 = 36 m, the width across the pond.
Key takeaways
Triangles are similar by AA (two pairs of congruent angles), SSS similarity (all three sides proportional), or SAS similarity (two sides proportional with congruent included angles). Congruent triangles are similar with scale factor 1, but AA alone never proves congruence. Similar triangles allow indirect measurement, from shadows of trees and pyramids, as Thales is said to have done around 600 BCE, to distances across ponds. The altitude to a right triangle's hypotenuse creates three similar triangles.
Flashcard
Two pairs of congruent angles
Check your understanding
A 1.8 m person casts a 2.4 m shadow while a tree casts a 20 m shadow. How tall is the tree?