Deductive and Inductive Thinking in Geometry

# Deductive and Inductive Thinking in Geometry

In the consideration of shapes, conclusions about shapes can and need to be drawn. Using the knowledge of various properties about shapes a variety of conclusions can be developed. It is important to understand the criteria upon which a shape is defined. These criteria or definitions provide the basis upon which further conclusions can be drawn. For example, by understanding that quadrangles are shapes which have four sides, the conclusion that both rectangles and squares are quadrangles can be drawn.

The process of drawing conclusions about shapes and theories is an important aspect of geometry. It is necessary to understand completely the basic definitions or criteria which create a specific situation in order to be able to make appropriate conclusions. This process can be made easier with a complete understanding of terms such as:
• Angles
• Parallel
• Lines
• Line Segments
• Rays
• Polygons
• Triangles
• Parallelograms

There are numerous terms which could be covered in order to completely understand the concept of geometry, too many to cover in this lesson. What is most important is the process upon which conclusions can and should be drawn. Drawing conclusions requires the use of deductive thinking. There are two types of thinking: inductive and deductive.

In inductive thinking conclusions are determined by making observations. Inductive reasoning alone is not a valid method for proving something is truth. Even the best scientists who observe across a variety of settings over different time periods cannot claim that any patterns they have observed will apply to all situations. Scientists utilize inductive thinking often to develop hypotheses which would need to have significant further study before they could be considered to be true. Many of the known truths began through a process of inductive reasoning and were further studied to develop into proven statements. However, it is not through the process of inductive reasoning only which a truth can be identified. Inductive reasoning is important because it helps scientists to gather ideas about the world around us. Through direct observation, ideas and hypothesis statements are created which can then be explored to determine the truth or falsehood of these ideas. In geometry, inductive reasoning is the process of making generalizations about shapes and ideas based on observations. It is not the basis for proving any information to be true about a shape or object.

In deductive thinking, it is important to consider all of the information known to draw one generalization. In the example given, it is known that all quadrangles have four sides. It is also known that squares have four sides and that rectangles have four sides. Therefore, since both squares and rectangles meet the criteria of having four sides, they can also be considered a quadrangle.

Deductive thinking is the process of using information from previously known facts as the basis for a conclusion. It requires a step-by-step process of thinking in which each fact is considered individually first, then combined with the other known facts to arrive at one conclusion. As each new fact is considered, decisions need to be made as to whether or not the conclusion you are working with fits with each new piece of information. For example beginning with a general truth fact about a specific situation, such as all of the people in the room are male, a new fact presented such as Mary is in the room can immediately be discounted such that Mary is female not male and if all of the people in the room are male, Mary cannot be in the room. While this example is very simplified, it provides an understanding as to how the process of deductive thinking occurs. It is this type of thinking which allows for conclusions to be made in geometry and geometric proofs.

In relation to some universal truths about shapes, it is easiest to first consider conclusions which apply to all of a specific group of shapes or portions of shapes. Below are some conclusions which can be drawn about entire groups of shapes based on known information about the shapes:
• All parallelograms have congruent diagonals
• All quadrilaterals with congruent diagonals are parallelograms
• All prisms have a plane of symmetry
• All right prisms have a plane of symmetry
• All rectangles have 4 right angles
• All rhombus have opposite sides which are parallel
• All properties of trapezoids are also properties of isosceles trapezoids

Deductive thinking is not always as simple as these examples may make it seem. One situation which occurs is that conclusions are based upon ideas which are mistaken truths. In fact, the misuse of information is most often the cause of incorrect conclusions when reasoning in a deductive manner. In the previous example, perhaps Mary is the name of a male. Accepting the name of Mary to be female when in fact it belonged to a male would lead to an incorrect conclusion that Mary could not be in the room of only males, when in-fact he could be included in that room. For this reason, it is imperative to ensure that the ideas or truths on which all conclusions are based are completely and totally true. If they are false in any way, there can be no accurate conclusions drawn from the information. One incorrect conclusion can ruin the reliability of any further conclusions or ideas.

Another area where deductive thinking can break down is if one takes liberties with the truths. There is no room for the making of assumptions in the process of deductive reasoning. Returning to the previous example, even considering that Mary is male, that does not automatically mean that he would be located in the room. No where in the original statement did it indicate that all males in the universe were found in the room. Making the assumption that Mary is male and therefore must be found in the room is not necessarily true and could lead to conclusions being drawn which are incorrect.

Sometimes there are statements about numbers or objects which cannot be proven, but are accepted as truths, until such time as they are proven to be false. These accepted truths are referred to as axioms. Axioms are statements about numbers which are generally accepted as being a truth even though there is no definitive proof.