Mathematical propositions

In geometry, the process of reasoning is a principal way to discover properties of geometric figures. It would be instructive therefore to acquaint yourself with the forms of reasoning usual in geometry.

All facts established in geometry are expressed in the form of propositions. The propositionsthat we take for granted without proof are called assumptions. With regard to a different set of assumptions the same proposition may, or may not be true. The assumptions themselves are neither true nor false. They may be said to be “true” only in the sense that their truth has been assumed.

Definitions are propositions which explain what meaning one attributes to a name or expression.

Axioms (some axioms are traditionally called postulates) are those facts which are accepted without proof. This includes, for example, some propositions: through any two points there is a unique line; if two points of a line lie in a given plane then all points of this line lie in the same plane.

Propositions that can be logically deduced from the assumptions are often called theorems. For example, if one of the four angles formed by two intersecting lines turns out to be right, then the remaining three angles are right as well.

Corollaries are those propositions which follow directly from an axiom or a theorem. For instance, it follows from the axiom "there is only one line passing through two points" that "two lines can intersect at one point at most."

In any theorem one can distinguish two parts: the hypothesis and the conclusion. The hypothesis expresses what is considered given, the conclusion what is required to prove. For example, in the theorem "if central angles are congruent, then the corresponding arcs are congruent" the hypothesis is the first part of the theorem: "if central angles are congruent," and the conclusion is the second part: "then the corresponding arcs are congruent;" in other words, it is given (known to us) that the central angles are congruent, and it is required to prove that under this hypothesis the corresponding arcs are congruent.

It is useful to notice that any theorem can be rephrased in such a way that the hypothesis will begin with the word "if," and the conclusion with the word "then." For example, the theorem "vertical angles are congruent" can be rephrased this way: "if two angles are vertical, then they are congruent."

The theorem converse to a given theorem is obtained by replacing the hypothesis of the given theorem with the conclusion (or some part of the conclusion), and the conclusion with the hypothesis (or some part of the hypothesis) of the given theorem. For instance, the following two theorems are converse to each other:

If we call one of these theorems direct, then the other one should be called converse.

In this example both theorems, the direct and the converse one, turn out to be true. This is not always the case. For example the theorem: "if two angles are vertical, then they are congruent" is true, but the converse statement: "if two angles are congruent, then they are vertical" is false.

Indeed, suppose that in some angle the bisector is drawn. It divides the angle into two smaller ones. These smaller angles are congruent to each other, but they are not vertical.