Geometry

Polygon Similarity

Big Picture

Have you ever seen siblings who look so alike that one seems to be a miniature copy of the other? Similar polygons follow the same concept of being the same shape but different in size. We can prove similarity in polygons, and there are several ways to prove that triangles are similar. Similar polygons also have many properties and relationships that can be used to solve problems.

Key Terms

Similar Polygons: Two polygons with the same shape but not the same size.

Scale Factor: The ratio relating corresponding side lengths of two similar polygons.

Similar Polygons

Similar Polygons

Two polygons are similar if and only if:

  • They have the same number of sides
  • Corresponding angles are congruent
  • Corresponding lengths are proportional
         a. For similar triangles, corresponding lengths include side lengths, altitudes, medians, and midsegments.

The symbol ~ means similar. Figure A ~ Figure B is a similarity statement. In similarity statements, corresponding vertices are listed in the same order.

Don’t confuse ~, =, and \(\underline{=}\)

In two similar polygons, the ratio relating any two corresponding lengths is equal to the scale factor.

  • Scale factor = Length of polygon 1 : Corresponding length of polygon 2
  • Two congruent figures are similar. Their scale factor is 1:1

If you have dificulty matching up corresponding lengths, redraw the diagrams so that the corresponding lengths are easier to identify

By definition, all the angles and sides of regular polygons (e.g. equilateral triangles, squares, etc.) are congruent. As a result, all regular polygons with the same number of sides are similar.

Proving Triangle Similarity

AA Similarity Postulate

Angle-Angle (AA) Similarity Postulate: Two triangles are similar if two angles in one triangle are congruent to two angles in the other.

AA Similarity Postulate

SSS Similarity Theorem

Side-Side-Side (SSS) SimilarityTheorem: Two triangles are similar if all corresponding sides are proportional

SSS Similarity Theorem
  • When using the SSS Similarity Theorem,compare the shortest sides, the longestsides, and the remaining sides

Geometry

Polygon Similarity Cont.

Proving Triangle Similarity (cont.)

SAS Similarity Theorem

Side-Angle-Side (SAS) Similarity Theorem: Two triangles are similar if both of the following are true:

  • Two sides in one triangle are proportional to two sides in another triangle
  • The included angle in the first triangle is congruent to the included angle in the second
SAS Similarity Theorem

Indirect Measurement

Lengths can be measured indirectly by setting up similar triangles.

  • First, find the scale factor between the triangles by setting up a ratio between two known corresponding sides.
  • Then calculate the unknown length using proportions.

Notes