Geometry

Types of Transformations

Transformations move and modify geometric shapes. There are several types of transformations that all transform figures in different ways. These transformations can be rigid (isometric) or non-rigid.

Types o f Reasoning

Reasoning is a fundamental part of geometric proofs. There are several types of reasoning, many of which we innately and naturally picked up. However, the different types of reasoning can be categorized and are useful for understanding geometry.

Triangle Relationships

Triangles are polygons with three sides and three angles. They are classified by their angles as well as by their sides. Like other polygons, triangles have two sets of angles: interior angles and exterior angles. Specifically, a triangle’s interior angles always add up to 180°, and the exterior angles add up to 360°.

Triangle Congruence

One important relationship between triangles that we can prove is congruence, meaning that two triangles are exactly the same shape and size. congruent triangles have certain properties that are the same, namely their angle measurements and side lengths. Geometric theorems and postulates dictate the different characteristics of congruent triangles.

Triangles and Quadrilaterals

Triangles and quadrilaterals are among the more basic and common polygons. Triangles always have interior angles sum to 180° while quadrilaterals always have interior angles sum to 360°.

Trapezoids and Kites

Not all quadrilaterals are parallelograms. Trapezoids and kites are two non-parallelograms with special properties.

Symmetry and Tessellations

Symmetry has a lot to do with transformations, since symmetric figures are often “immune” to some types of transformations. This means that some symmetric figures don’t change after certain transformations. An interesting application of transformations is tessellations. Tessellation is when a plane can be covered by one or more shapes without any overlaps or gaps.

Surface Area and Volume

Surface area and volume are two very fundamental properties of 3-dimensional shapes. Often times in geometry we will be asked to find the surface area or volume of a shape. The more simple shapes can be solved by using a general formula. More complex shapes will require us to apply our knowledge of one or several 2-dimensional shapes.

Spheres

Spheres can be thought of as 3-dimensional circles. While all the points of a circle are equally far from the center in a 2-dimensional space, the equidistant points of spheres are in 3-dimensional space.

Special Right Triangles

Some triangles, called special right triangles, have simple formulas for calculating their side lengths. These include isosceles right triangles and triangles with the measures of 30°, 60°, and 90°. By memorizing these special right triangles, we can solve certain geometry problems faster.

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