similar triangles and proportional reasoning
What Are Similar Triangles?
Two triangles are similar if they have the same shape but not necessarily the same size. Mathematically, this means:
- Their corresponding angles are congruent (equal)
- Their corresponding sides are proportional
We denote similarity between triangles △ABC and △DEF as: △ABC ~ △DEF
When two triangles are similar, the ratio of any pair of corresponding sides is equal to the ratio of any other pair of corresponding sides. This constant ratio is called the scale factor.
Criteria for Similar Triangles
There are three main criteria to prove that triangles are similar:
- Angle-Angle (AA) Similarity: If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar. (Note: Since the sum of angles in any triangle is 180°, if two angles are equal, the third must also be equal.)
- Side-Side-Side (SSS) Similarity: If the corresponding sides of two triangles are proportional, then the triangles are similar.
- Side-Angle-Side (SAS) Similarity: If two sides of one triangle are proportional to two sides of another triangle, and the included angles are congruent, then the triangles are similar.
Properties of Similar Triangles
When triangles are similar, several important relationships exist:
- The ratio of corresponding sides equals the scale factor (k)
- The ratio of perimeters equals the scale factor (k)
- The ratio of areas equals the square of the scale factor (k²)
- The ratio of altitudes to corresponding sides equals the scale factor (k)