Geometry is a wonderful part of mathematics for people who don't like a lot of numbers. It has shapes and angles, and it also has logic. Logic is formal, correct thinking, reasoning, and inference. Logic is not something humans are born with; we have to learn it, and geometry is a great way to learn to be logical.
You may know the word converse for a verb meaning to chat, or for a noun as a particular brand of footwear. Neither of those is how mathematicians use converse. Converse and inverse are connected concepts in making conditional statements.
To create the converse of a conditional statement, switch the hypothesis and conclusion. To create the inverse of a conditional statement, turn both hypothesis and conclusion to the negative.
Conditional statements set up conditions that could be true or false. These conditions lead to a result that may or may not be true. Conditional statements start with a hypothesis and end with a conclusion.
Here are examples of conditional statements with false hypotheses:
You can test the hypothesis immediately: Are you 9 meters tall? Do squares have three sides? These conditional statements result in false conclusions because they started with false hypotheses.
Conditional statements begin with "If" to introduce the hypothesis. The hypothesis is the part that sets up the condition leading to a conclusion. The conclusion begins with "then," like this:
You will see conditional statements in geometry all the time. You can set up your own conditional statements. Here is one for an isosceles triangle:
You can switch the hypothesis and conclusion of a conditional statement. You take the conclusion and make it the beginning, and take the hypothesis and make it the end:
The converse of a true conditional statement does not automatically produce another true statement. It might create a true statement, or it could create nonsense:
That statement is true. But the converse of that is nonsense:
We know it is untrue because plenty of quadrilaterals exist that are not squares.
Many times in geometry we see postulates and theorems that seem like they could become conditional statements and converse conditional statements:
Some postulates are even written as conditional statements:
Below we have equilateral triangle . We can set up conditional statements about it. Here are five statements. Decide which ones are conditional, which are not conditional, and which conditional statements are true:
Statements 1, 2, and 5 are all true conditional statements (If … then).
Statement 3 is a converse of statement 2.
Statement 4 is not a conditional statement, but it is true. You have enough information to change statement 4 into a conditional statement.
Let's check the converse statement, 3, to see if it is true. Can you create a triangle with one interior angle measuring but with the other angles having different measures? Of course you can, like a triangle, which is definitely not equilateral. So the converse statement is not true.
In this lesson you have learned to identify and explain conditional statements and create your own conditional statements. You know conditional statements could be true or false. You are able to exchange the hypothesis and conclusion of a conditional statement to produce a converse of the statement, and you can test to see if the converse of a true conditional statement is true.
After checking out the multimedia and these directions, you will be able to:
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