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# Geometry POLYGONS

Triangles, with their three sides and three angles, stand as the fundamental building blocks of two-dimensional geometry. Their simplicity makes them a natural starting point for understanding the principles of shape and space. However, as we move on to more complicated shapes, we start looking at polygons which can have any number of sides.  Believe it or not, there’s even such a thing as a 10 sided polygon! This downloadable study guide primarily covers the classification of polygons, the concavity and convexity of geometric shapes, and different types of 4 sided shapes called quadrilaterals.  All math classes are cumulative, so remembering earlier theorems about parallel and perpendicular lines will be important to do well in this new unit.  If you have a B- or below in your course at this point, it’s time for a geometry tutor.

It’s always important for you to learn to construct geometric shapes on your own, and it helps to see them rather than just read about them, but here are some ideas to get you started.

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In general, polygons are named for the number of sides they have.  Pentagons have five sides, octagons have eight sides, and decagons have 10 sides. The decagon, like all polygons, can take various forms, each with its own set of properties. One of it’s obvious features is that it has ten angles and ten vertices. However, beyond this basic definition, the properties of a decagon can vary based on whether it is regular or irregular.  A regular decagon is characterized by all sides and angles being equal. Achieving this uniformity imparts a sense of symmetry and order to the shape. Regular polygons, in general, are often associated with being aesthetically pleasing. On the other hand, an irregular decagon has sides and angles of varying lengths and measures, resulting in a less uniform appearance.

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Quadrilaterals, as the name suggests, are polygons with four sides. They are a diverse family of shapes that includes squares, rectangles, parallelograms, rhombuses, kites and trapezoids, each with its unique set of properties and characteristics.  Which categories a quadrilateral falls into depends on the number of parallel or perpendicular lines in the shape, how many sides are congruent, and how many angles are congruent.

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A hexagon, by definition, is a polygon with six sides. Now, depending on its shape, a hexagon can be either convex or concave.  To determine which of the two it is,  we must first clarify the concepts of concavity and convexity in geometry. A polygon is considered convex if when the lines that make up it's sides are extended and they intersect outside of the shape.   A regular hexagon, where all sides and angles are equal, is an example of a convex hexagon, and all the interior angles are all less than 180 degrees. On the other hand, a hexagon is concave if it has at least one interior angle greater than 180 degrees, causing it to "bend inward," creating a concave shape. When the lines that make up its sides are extended,  they intersect inside of the shape. This distinction highlights the flexibility and diversity within the family of hexagons, demonstrating how geometric properties can vary based on the specific characteristics of a shape.

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