• Basic Geometric Figures...Figures created by geometry can be intricately complex. However, they may be broken down into simpler parts. The most fundamental components are points, and the lines between points. In fact, these concepts are so simple and basic that points and lines are thought of as undefined. A point has no size or shape. A line has no thickness. Therefore they occupy no real space, and so we cannot define them in the way that we define a real object. Lines, rays, planes, and angles extend infinitely, another abstraction, which is unlike real objects. By studying and exploring these concepts, we may define the characteristics of real-life objects. That is why geometry is one of the most ancient and culturally widespread disciplines of mathematics. Whether you want to plant a field with crops or build a house, it all breaks down into points and lines.
• Basic Geometric Properties...Points, planes, lines (parallel and perpendicular) are the basic building blocks of geometry. They are very important concepts that we use to understand more advance topics in geometry. Everything starts with a point; a simple immeasurable spot in space. A string of points forms lines and line segments. A geometric plane is a surface that extends forever where points can be plotted to draw parallel lines that never cross and perpendicular lines that cross at 90-degree angles.
• What's Your Angle?...In this module, students will learn about the angles of a triangle, regular polygon, and irregular polygons. These geometry subjects are vital to understand and are the building blocks to more advanced geometric angles.
Definitions, Postulates, Theorems
• Introduction to Theorems...Deductive Reasoning, If-Then Statements, Converses, Postulates and Theorems that Relate Points/Lines/Planes This module discusses the difference between inductive and deductive reasoning, the relationship of deductive reasoning to geometry, basic techniques of deductive reasoning, and postulates and theorems related to points, lines, and planes.
• Where's the Proof?...In this module we will discuss how to prove theorems, how to plan a proof, special pairs of angles, theorems about angles and perpendicular lines, and proving that lines are parallel.
• Comparing Attributes of Triangles...In your studies of mathematics you have encountered many laws that govern operations. Oftentimes, the law is quoted as a rule without an explanation of why the rule is true. The study of congruent triangles opens the door on the laws of math, illustrating how it all works. Obviously, this is useful knowledge to have for solving geometry problems; however, geometric proofs can be applied to algebra and arithmetic. They historically have been the basis for much of our mathematical knowledge. The concept of a proof, or how we know what we know, has applications beyond pure math. It is how all manner of logic may be rendered into a systematic process and how philosophical arguments are analyzed. The methods for proving congruence in basic triangles are an easy-to-comprehend way of leaning how a logical proof is constructed.
• A Closer Look Within the Triangle...The principles shown in this lesson are examples of how familiar concepts, such as the height of a triangle, may be expanded into broader concepts such as altitude. These are only a few of the fundamental possibilities for examining the characteristics of triangles and other polygons. For every line or angle, there are a wide variety of connecting lines that may be drawn to convey the mathematical relationships within the triangle.
• Properties of Quadrilaterals...The properties of parallel lines may be applied to quadrilaterals, or four-sided figures. Polygons made from two sets of parallel lines are called parallelograms. Parallelograms are convenient to use because they allow us to apply the characteristics of parallel lines to representations of real-life objects. Parallel lines share mathematical data, such as slope or angle. After finding the data relevant to one line, we can find similar data for lines parallel to the first line. When this principle is applied to a shape, such as a polygon, the information becomes a useful way of making calculations. For example, the parts of a building, such as bricks or wooden beams, fit together one on top of the other to form parallel lines. This means that that architects and builders must often find parallel lines and that the study of parallel lines has many applications.
• Special Geometric Figures...This module defines and covers the properties of parallelograms, including rhombi, rectangles, and squares, trapezoids, and kites. Students will learn how the quadrilaterals are related and how they differ. The study on trapezoids includes isosceles trapezoids and medians (mid-segments).
• Exploring Inqualities in Geometry...The uses of inequalities in geometry are discussed, as well as several methods of negative proofs.
• Comparing Polygons...Ratio and proportion are familiar concepts from arithmetic and algebra. Applied to geometry, they allow us compare objects of different sizes but similar shapes. The component segments of a given figure may be compared to other segments of the figure, allowing us to define its characteristics. An architect can draw a blueprint that is only a few feet across but that represents a large building. The drawing could even be revised to show buildings of different sizes but similar proportions. In the past, engineers often used scales in multiples of 12 and 8, based upon the divisions of an inch ruler. An inch on a 1:12 plan could be instantly converted to 1 foot when working with the actual object; likewise, a 1/8-inch hatch-mark on a 1:96 plan could be instantly known to represent a foot.
• Triangle Theorems...The rules which govern triangles are explained, such as how to derive several measurements when one measurement is known.
• Introduction to Trigonometric Functions...The Sine, Cosine, and Tangent Ratios, Right Triangle Trigonometry Trigonometry is an outgrowth of the math of similar triangles. This module introduces students to the basic definitions, SOH-CAH-TOA formulas, and some applications of trigonometry of the right triangle.
• Inside the Circle of Trust...Studying the intersection of lines and circles is a useful way of finding the characteristics of both. After learning the fundamentals of this interaction, we can find ways to apply that knowledge to everyday life. The number of revolutions required by gears in a machine, the composition of subjects in a photograph, and the design of a building are all examples of arc measurement. Additionally, the basic concepts of circles will prepare you to learn the more advanced operations of trigonometry.
Locus and Constructions
• Time to Draw...Drawing techniques for geometric constructions with a compass and straightedge are explained with examples and exercises.
• Focus on the Locus...The concept of locus is introduced, including examples for lines, circles, parabolas, and ellipses.
• Investigating Polygons...Formulas used to find the area of polygons and circles are explained and discussed, as well as arc length, radians, and ratios of area.
Geometry and Algebra
Coordinate Geometry and Transformations
• The Precise Coordinates...The application of linear algebra to geometry is explained, especially in the context of proofs. This includes finding the equations and slope for a line, as well as using slope to identify perpendicular or parallel lines.
• Manipulations of Geometric Figures...Understanding geometric transformations is key to understanding how geometry is applied in the real world. Computer animations, architecture, and mechanics all involve figures that move or must fit exactly together. Learning how to use these geometric principles is an excellent way to prepare for careers in a variety of disciplines.