Many postulates and theorems in geometry are conditional statements. Let's find out what that means.
The following statements are examples of conditional statements.
- It is raining, so you will need an umbrella.
- Get three months free when you sign up for the gym.
- Free phone with a two-year service contract.
That means they can be rewritten in if-then form.
- If it is raining, then you will need an umbrella.
- If you sign up for the gym, then you will get three months free.
- If you sign a two-year service contract, then you will receive a free phone.
The phrase that follows the word "if" is called the hypothesis of the statement, and the phrase following the word "then" is called the conclusion.
Now you try! Write the following statements in "If. . . then. . ." form.
- Cows eat grass.
- Three points are collinear if they lie on the same line.
- The sum of the angles in a triangle total 180 degrees.
Click the link below to complete the activity.Conditional Statements Activity
In the next section, you will explore some other types of statements that we use in geometry.
More Conditional Statements
More Conditional Statements
In the last activity, you practiced writing conditional statements, or "If. . . then. . ." statements. In this section, we are going to learn about three other types of statements you will encounter in geometry.
- If you exchange the hypothesis and the conclusion of a conditional statement, it is called the converse.
- If you negate both the hypothesis and the conclusion of a conditional statement, then you form the inverse of the statement.
- To form the contrapositive of a statement, you negate both the hypothesis and the conclusion of the converse statement.
Just because a conditional statement is true, it does not guarantee that the converse and inverse are also true. The contrapositive of a true conditional statement is always true, just like the contrapositive of a false conditional statement is always false.
There is a fourth type of statement called the biconditional statement. If a conditional statement and its converse are both true, the biconditional statement can be written using the phrase "if and only if."
Now, it is your turn to choose the conditional, converse, inverse, contrapositive, and biconditional for some statements in the activity below.Conditional Statements Activity
The next activity will provide an opportunity to explore the validity of statements.
It may seem simple to write the converse, inverse, contrapositive, or biconditional statements related to a conditional statement, but are all of the statements true?
Let's take a look at the following example.
If a quadrilateral is a rectangle, the diagonals have the same length.
State the contrapositive and determine whether it is valid.
To state the contrapositive, we will need to first write the converse and then negate both the hypothesis and the conclusion.
|Converse:||If the diagonals of a quadrilateral have the same length, then it is a rectangle.|
|Contrapositive:||If the diagonals of a quadrilateral do not have the same length, then it is not a rectangle.|
To determine if the statement is valid, we need to consider if there are any counterexamples. Ask yourself if a rectangle will ever have diagonals of unequal length? It will not, so this statement is true.
Now try this one on your own.Statement Validity Activity
You will continue the use of conditional statements, their converses, inverses, and contrapositives throughout the study of geometry.