Remember how to write a converse? Given a conditional statement, such as if P then Q, we can write the converse statement as If Q, then P.
How would you write the converse of the Triangle Inequality Theorem? The theorem states that in a triangle, any side must be shorter than the sum of the other two sides.
To construct the converse of the theorem, let's assign statements to the variables P and Q. If we rewrite the theorem in conditional form, we get: If a triangle exists, then the sum of any two sides will be greater than the third. These P and Q of this conditional statement are as follows:
P = a triangle exists
Q = any side must be shorter than the sum of the other two sides
Question
Given the definition of converse, what would the converse of the Triangle Inequality Theorem be?
If you reverse P and Q, you get this conditional statemeent: In a three-sided shape, if any side is shorter than the sum of the other two sides, then a triangle exists.
Once you know the theorem's converse, how can you use it? If you think about this statement for a moment, you can see that it's actually a more intuitive way to determine if you have a triangle when given the lengths of three sides. If you picture a triangle in your mind, visually speaking, any side would need to be smaller than the other two sides put together. Otherwise, it would be impossible to form the triangle.
To put this visual rule into practice, consider each of the photos below. When you visually inspect an object to see if it follows the Triangle Inequality Theorem, there is no real need to label the triangle or write down any inequalities.
