Rube Goldberg was a cartoonist in the 1940s who drew crazy inventions to do very simple things. The invention below has a series of smaller tasks that leads to the machine wiping the man’s face with a napkin.

Write a series of if-then statements to that would caption this cartoon, from \begin{align*}A\end{align*} to \begin{align*}M\end{align*}.

### Conditional Statements

A **conditional statement** (also called an **If-Then Statement**) is a statement with a hypothesis followed by a conclusion. Another way to define a conditional statement is to say, “If this happens, then that will happen.”

The **hypothesis** is the first, or “if,” part of a conditional statement. The **conclusion** is the second, or “then,” part of a conditional statement. The conclusion is the result of a hypothesis. Keep in mind that conditional statements might not always be written in the “if-then” form. Here are a few examples.

**Statement 1:**If you work overtime, then you’ll be paid time-and-a-half.

**Statement 2:**I’ll wash the car if the weather is nice.

**Statement 3:**If 2 divides evenly into \begin{align*}x\end{align*}, then \begin{align*}x\end{align*} is an even number.

**Statement 4:**I’ll be a millionaire when I win monopoly.

**Statement 5:**All equiangular triangles are equilateral.

Statements 1 and 3 are written in the “if-then” form. The hypothesis of Statement 1 is “you work overtime.” The conclusion is “you’ll be paid time-and-a-half.” So, if Sarah works overtime, then what will happen? From Statement 1, we can conclude that she will be paid time-and-a-half. If 2 goes evenly into 16, what can you conclude? From Statement 3, we know that 16 must be an even number. Statement 2 has the hypothesis after the conclusion. Even though the word “then” is not there, the statement can be rewritten as: If the weather is nice, then I’ll wash the car. If the word “if” is in the middle of a conditional statement, the hypothesis is always after it. Statement 4 uses the word “when” instead of “if.” It should be treated like Statement 2, so it can be written as: If I win monopoly, then I will be a millionaire. In Statement 5 “if” and “then” are not there, but can be rewritten as: If a triangle is equiangular, then it is equilateral.

#### Rewriting Statements in If-Then Form

Rewrite the following statement in if-then form: *All students like geometry.*

Rewritten in if-then form, this statement would be *if you are a student, then you like geometry*.

#### Identifying Hypotheses and Conclusions

Identify the hypothesis and the conclusion of the statement: Bob will go to the store if Anne tells him what to buy.

First, rewrite in if-then form. *If Anne tells Bob what to buy, then Bob will go to the store*. The hypothesis is *Anne tells Bob what to buy* because this has to come first. The conclusion, or result, is *Bob will go to the store.*

#### Rewriting a Statement in If-Then Form

Identify the hypothesis and the conclusion of the statement: *I bring my umbrella when it is raining*.

Rewrite in if-then form, considering what causes what. In this situation, it is the rain that causes me to bring an umbrella (not bringing an umbrella that causes rain). *If it is raining, then I bring my umbrella*. The hypothesis is *it is raining* and the conclusion is *I bring my umbrella*.

#### Earlier Example Revisited

The following answers apply to the question asked at the beginning of this Section. The conditional statements are as follows:

\begin{align*}A \rightarrow B\end{align*}: If the man raises his spoon, then it pulls a string.

\begin{align*}B \rightarrow C\end{align*}: If the string is pulled, then it tugs back a spoon.

\begin{align*}C \rightarrow D\end{align*}: If the spoon is tugged back, then it throws a cracker into the air.

\begin{align*}D \rightarrow E\end{align*}: If the cracker is tossed into the air, the bird will eat it.

\begin{align*}E \rightarrow F\end{align*}: If the bird eats the cracker, then it turns the pedestal.

\begin{align*}F \rightarrow G\end{align*}: If the bird turns the pedestal, then the water tips over.

\begin{align*}G \rightarrow H\end{align*}: If the water tips over, it goes into the bucket.

\begin{align*}H \rightarrow I\end{align*}: If the water goes into the bucket, then it pulls down the string.

\begin{align*}I \rightarrow J\end{align*}: If the bucket pulls down the string, then the string opens the box.

\begin{align*}J \rightarrow K\end{align*}: If the box is opened, then a fire lights the rocket.

\begin{align*}K \rightarrow L\end{align*}: If the rocket is lit, then the hook pulls a string.

\begin{align*}L \rightarrow M\end{align*}: If the hook pulls the string, then the man’s faces is wiped with the napkin.

### Examples

First rewrite in if-then form, then determine the hypothesis and conclusion.

#### Example 1

Sally eats a snack if she is hungry.

In if-then form, the statement is *If Sally is hungry, then she eats a snack. *The hypothesis is *Sally is hungry *and the conclusion is *she eats a snack.*

#### Example 2

The angles in a triangle add up to 180 degrees.

In if-then form, the statement is *If a shape is a triangle, then its angles add up to 180 degrees*. The hypothesis is *a shape is a triangle* and the conclusion is *its angles add up to 180 degrees*.

#### Example 3

2012 is a leap year.

In if-then form, the statement is *If it is 2012, then it is a leap year.* The hypothesis is *it is 2012* and the conclusion is *it is a leap year.*

### Review

For questions 1-10, determine the hypothesis and the conclusion.

- If 5 divides evenly into \begin{align*}x\end{align*}, then \begin{align*}x\end{align*} ends in 0 or 5.
- If a triangle has three congruent sides, it is an equilateral triangle.
- Three points are coplanar if they all lie in the same plane.
- If \begin{align*}x =3\end{align*}, then \begin{align*}x^2=9\end{align*}.
- If you take yoga, then you are relaxed.
- All baseball players wear hats.
- I'll learn how to drive when I am 16 years old.
- If you do your homework, then you can watch TV.
- Alternate interior angles are congruent if lines are parallel.
- All kids like ice cream.

Rewrite each statement in if-then form.

- Susie eats pizza every Thursday.
- Raychel always completes her homework.
- Alex goes to school every weekday.
- All students have a math class.
- Squares have right angles.

### Review (Answers)

To view the Review answers, open this PDF file and look for section 2.2.