Conditional Statements

Logic is a big part of geometry and although one does not need to know formal logic in order to do geometry, it helps to start geometry by starting to think about logic and how it works.  Geometry logic starts with conditional statements.

Conditional statements are "if-then" statements.  The "if" part of the statement is called the hypothesis.  The "then" part of the statement is the conclusion.  All conditionals contain a hypothesis and conclusion, even if they do not explicitly contain the words "if" and "then."

Example:

If two lines do not intersect, they are parallel.

Here, the hypothesis is "If two lines intersect" and the conclusion is "they are parallel."

Conditional statements get their power from being always true.  So, the way to prove a conditional wrong is to prove a counterexample. A counterexample is a single example that disproves the conditional statement.

Example:

If a figure has four sides, it is a square. 

Counterexample: a rectangle that does not have equal sides.  This shape has four sides but is not a square.  This conditional is false.

Every condition has a converse.  Converse statements switch the hypothesis and the conclusion. If a conditional is true, sometimes the converse is true and sometimes it's not. 

Examples: 

Conditional: If two lines do not intersect, they are parallel.

Converse: If two lines are parallel, they do not intersect.

Both the conditional and the converse are true. 

Conditional: If $x=1$ and $y=2$, then $x+y=3$.

Converse: If $x+y=3$, then $x=1$, and $y=2$.

Counterexample: $x=3$ and $y=0$.

The conditional is true, but the converse is not true. 

For every conditional statement you can also have an inverse statement.  An inverse of a conditional is formed by providing the negative form of both the hypothesis and the conclusion. 

Examples: 

Conditional: If two lines do not intersect, they are parallel. 

Inverse: If two lines do intersect, they are not parallel. 

Both the conditional and the inverse are true.

Conditional: If an angle is greater than 90 degrees, it is obtuse. 

Inverse: If an angle is not greater than 90 degrees, it is not obtuse. 

Both the conditional and the inverse are true. 

For every conditional statement, you can also have a contrapositive statement.  A contrapositive statement is formed by providing the negative of both the hypothesis and the conclusion of the CONVERSE of a statement. 

Examples:

Conditional: If two lines do not intersect, they are parallel. 

Contrapositive: If two lines are not parallel, they intersect. 

Both the conditional and the contrapositive are true.

Conditional: If an angle is greater than 90 degrees, it is obtuse. 

Inverse: If an angle is not obtuse, it is not greater than 90 degrees. 

Both the conditional and the contrapositive are true.