Geometry is a language that discusses shapes and angles blended in algebraic terms. Geometry expresses the relationships between one-dimensional, two-dimensional and three-dimensional figures in mathematical equations. Geometry is used extensively in engineering, physics and other scientific fields. Students gain insight into complex scientific and mathematical studies by learning how geometric concepts are discovered, reasoned and proved.
Inductive reasoning is a form of reasoning that arrives at a conclusion based on patterns and observations. If used by itself, inductive reasoning is not an accurate method for arriving at true and accurate conclusions. Take the example of three friends: Jim, Mary and Frank. Frank observes Jim and Mary fighting. Frank observes Jim and Mary argue three or four times during the week, and each time he sees them, they are arguing. The statement, “Jim and Mary fight all the time,” is an inductive conclusion, reached by limited observation of how Jim and Mary interact. Inductive reasoning can lead students in the direction of forming a valid hypothesis, such as “Jim and Mary Fight often.” But inductive reasoning cannot be used as the sole basis to prove an idea. Inductive reasoning requires observation, analysis, inference (looking for a pattern) and confirming the observation through further testing to arrive at valid conclusions.
Deductive reasoning is a step-by-step, logical approach to proving an idea by observation and testing. The deductive reasoning starts with an initial, proven fact and builds an argument one statement at a time to undeniably prove a new idea. A conclusion arrived at through deductive reasoning is built on a foundation of smaller conclusions that each progress toward a final statement.
Axioms and Postulates
Axioms and postulates are used in the process of developing inductive- and deductive-reasoning arguments. An axiom is a statement about real numbers that is accepted as true without requiring a formal proof. For example, the axiom that the number three possesses a larger value than the number two is a self-evident axiom. A postulate is similar, and defined as a statement about geometry that is accepted as true without proof. For example, a circle is a geometric figure that can be divided evenly into 360 degrees. This statement applies to every circle, in all circumstances. Therefore, this statement is a geometric postulate.
A theorem is the result or conclusion of an accurately built deductive argument, and can be the result of a well-researched inductive argument. In short, a theorem is statement in geometry that has been proved, and therefore can be relied upon as a true statement when building logical proofs for other geometry problems. The statements that “two points determine a line” and “three points determine a plane” are each geometric theorems.