# If-Then Statements

## Hypothesis followed by a conclusion.

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If-Then Statements

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.

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

Rewrite each statement in if-then form.

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

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Color Highlighted Text Notes

### Vocabulary Language: English

TermDefinition
Angle A geometric figure formed by two rays that connect at a single point or vertex.
Subset A subset is a collection of numbers or objects within a larger set.