The theoretical aspect of geometry is composed of definitions and theorems. Definitions are precise descriptions of words used in geometry. Theorems are logical results deduced about these geometric definitions. On this page, we will discuss precise definitions.

bullet What do we know about definitions?

definition A definition is the formal statement of the meaning or significance of a word.

Definitions are bi-directional. If you put a definition into "if - then" form, the definition will be true when the portions of the "if" and "then" statements are exchanged. Both the definition and its converse are true. Logically speaking, this is referred to as an "if and only if" situation, or a biconditional.

Definition: A right triangle is a triangle with one right angle.

If-then form:  If a triangle is a right triangle, then it is a triangle with one right angle. (true)
Converse (Reversed form):   If a triangle has one right angle, then it is a right triangle. (true)
Biconditional:  A triangle is a right triangle if and only if the triangle has one right angle.

bullet What do we need to know about mathematical definitions?

A good mathematical definition explains precisely what something means. You may express definitions in your own words, as long as your definitions are precise.

While expressing definitions in your own words sounds like a simple process, it can easily lead to faulty definitions and loss of credit on test papers. Be careful!

Take a look at what problems arise when precision is lacking.
Student's Definition: Complementary angles are angles adding to 90 degrees.

(Note: This definition is considered incorrect.)

Counter-example to disprove student definition:
By the student definition, ∠1, ∠2 and ∠3 could be complementary.
This is not correct!
Precise Definition:
Complementary angles are TWO angles the sum of whose measures is 90.

If you choose to define terms "in your own words", be sure to express a degree of precision that will guarantee your definition is accurate.


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