Similar polygons and triangles are all about shape twins. They look alike but can be different sizes. The key is that their angles match up, and their sides are in proportion. This concept is super useful for solving real-world problems.

You can tell if shapes are similar by checking their angles or sides. Once you know they're similar, you can use ratios to figure out missing measurements. This comes in handy for things like indirect measurement and finding geometric means in right triangles.

Similar Polygons and Triangles

Definition of similar polygons

• Similar polygons have congruent corresponding angles and proportional corresponding sides
• Denoted using the similarity symbol $\sim$ (triangles $\triangle ABC \sim \triangle DEF$)
• Examples of similar polygons include equilateral triangles, squares, and regular pentagons of different sizes

Conditions for polygon similarity

• AA (Angle-Angle) Similarity Theorem states that if two angles of one triangle are congruent to two angles of another triangle, the triangles are similar
• SAS (Side-Angle-Side) Similarity Theorem states that if the ratio of the lengths of two corresponding sides of two triangles is equal to the ratio of the lengths of two other corresponding sides, and the included angles are congruent, the triangles are similar
• SSS (Side-Side-Side) Similarity Theorem states that if the ratios of the lengths of the corresponding sides of two triangles are equal, the triangles are similar
• Two polygons are similar if they have the same number of sides, corresponding angles are congruent, and corresponding sides are proportional (regular hexagons, pentagons)

Calculations in similar shapes

• In similar polygons or triangles, the ratio of the lengths of any two corresponding sides is equal to the scale factor
  • $\frac{side1_{polygon1}}{side1_{polygon2}} = \frac{side2_{polygon1}}{side2_{polygon2}} = \frac{side3_{polygon1}}{side3_{polygon2}} = ... = scale factor$
• Corresponding angles in similar polygons or triangles are congruent
  • $\angle A \cong \angle D, \angle B \cong \angle E, \angle C \cong \angle F$ (triangles)
• The ratio of the perimeters of two similar polygons is equal to the scale factor
• The ratio of the areas of two similar polygons is equal to the square of the scale factor

Problem-solving with similar polygons

1. Indirect measurement uses similar triangles to find the height of tall objects (buildings, trees) or the distance across wide spaces (rivers, canyons)

   • Set up a proportion using the corresponding sides of the similar triangles

2. In a right triangle, the geometric mean states that the length of the altitude to the hypotenuse is the geometric mean of the lengths of the two segments of the hypotenuse

   • $\frac{a}{x} = \frac{x}{b}$, where $a$ and $b$ are the lengths of the segments of the hypotenuse, and $x$ is the length of the altitude

3. Solve for missing side lengths or angle measures in similar polygons using proportions (scale factor) and the properties of similar polygons (congruent angles, proportional sides)