Section 1 - Theorems

The majority of geometric ideas are expressed in the form of a theorem. A theorem is a statement that includes a hypothesis and a conclusion, which are written as a sentence using "if" and "then". For example:

If a gem is green, then it is an emerald.

"If a gem is green" is the hypothesis, and "then it is an emerald" is the conclusion. All theorems also have a converse. The converse of a theorem is written by switching the hypothesis and conclusion around. For example:

If it is an emerald, then it is a green gem.

This theorem is the converse of the first example. Not all theorems are true, either.

If it gives milk, then it is a cow.

This is not true, because many other animals give milk. If a theorem is to be wrong, then you must write a counterexample. A counterexample is a statement that gives an example that follows the hypothesi, yet does not follow the conclusion.

Theorem: If it gives milk, then it is a cow.

Counterexample: Goats also give milk.

Not too tough, is it?

Practice away!

Write the hypothesis of the following theorems-

1) If a man is 7 feet tall, then he plays basketball.

2) If you are flying, then you are in a jet.

3) If you are in Alaska, then you are cold.

Write the conclusion of the following theorems-

4) If you are 15 years old, then you are a freshman.

5) If you are not in school, then it is summer.

6) If an animal does tricks, then it is a dog.

Write the converse of the following theorems-

7) If it is a ring, then it is made of gold.

8) If it is a bird, then it can fly.

9) If you wear diapers, then you are an infant.

10) If you play football, the you wear a uniform.

Decide whether each theorem is true or false, if false, write a counterexample.

11) If it is a battery, then it is rechargable.

12) If it is a computer, then it has a hard drive.

13) If an animal is black and white, then it is a penguin.

14) If an animal walks on two feet, then it is a human.

15) If a gem is a ruby, then it is red.

16) If it is a fish, then it lives in water.