A statement joining two events together based on a condition in the form of “If something, then something” is called a conditional statement.

In Geometry, conditional statements, which are also called “If-Then” statements, are written in the form:

If p, then q. 

Mathematicians will also use logic symbols to indicate a conditional statement, using an arrow to replace the words “if” and “then.” The sentence below is read “p implies q.”

p → q

Either of these structures have the same result:

If the first event occurs, then the second event will result as a condition of the first event.

Inquiring Minds Want to Know: Why do we care about conditional statements?

Conditional statements imply a cause-and-effect relationship, and can be tremendously useful in Geometry. Mathematicians and scientists like conditional statements because of the relationship between the two events. They know that if they see the first event happening, then the second event will follow. Knowing about conditional statements enables mathematicians and scientists to make predictions about the future!

Let’s begin by examining some structures for conditional statements with real-world examples. Don’t worry about the truth value of those statements for now. We will look into how to determine the truth value of a conditional statement in a later lesson.

Examples:

Fill in the blanks.

• If you get a 100 on your final exam, then your teacher will give you an “A”
  • The “If” statement or p is Interactive button. Assistance may be required. ____________. you get a 100 on your final exam
  • The “then” statement or q is Interactive button. Assistance may be required. __________. your teacher will give you an “A”
    
    
• If you live in Austin, then you live in Texas.
  • The “If” statement or p is Interactive button. Assistance may be required. ____________. you live in Austin
  • The “then” statement or q is Interactive button. Assistance may be required. __________. you live in Texas
    
    
• If a number is divisible by 10, then the number ends in zero.
  • The “If” statement or p is Interactive button. Assistance may be required. ____________. a number is divisible by 10
  • The “then” statement or q is Interactive button. Assistance may be required. __________. the number ends in zero

Conditional statements are so important because they express a relationship between two events. There are several ways to describe the two events linked in a conditional statement.

• The Hypothesis is the first or “If” portion of the conditional statement or p.
• The Conclusion is the second or “then” part of the conditional statement or q.

Important Notes:

• The word “IF” is not part of the hypothesis.
• The word “THEN” is not part of the conclusion.
• The words “IF” and “THEN” create the relationship using a sentence structure.

We can use a graphic organizer to show how If-then, Hypothesis-Conclusion, and p → q notation are related. Drag the terms into the appropriate place within the graphic organizer.

You may want to record the summarizing graphic organizer using your notes.

In each of the following examples, identify both the Hypothesis and the Conclusion:

• If you have a coupon, then you will receive a free drink with your pizza.
• Hypothesis:  Interactive popup. Assistance may be required. Check Your Answer You have a coupon.
• Conclusion: Interactive popup. Assistance may be required. Check Your Answer You will receive a free drink with your pizza.

• If there is no school, then it is the weekend.
• Hypothesis:  Interactive popup. Assistance may be required. Check Your Answer There is no school.
• Conclusion: Interactive popup. Assistance may be required. Check Your Answer It is the weekend.

Not every cause-effect relationship is presented in If-then or conditional form. Sometimes, you have to interpret the statement to determine the hypothesis and conclusion before you can write the statement in conditional form.

Examples: Write the conditional statements for the following relationships.

• You are not in school during the summer.
• Hypothesis: Interactive popup. Assistance may be required. Check Your Answer You are not in school.
• Conclusion: Check Your Answer It is summer.
• Conditional Statement: Interactive popup. Assistance may be required. Check Your Answer If you are not in school, then it is summer.

• The Tigers will play in the finals after winning this game.
• Hypothesis:  Interactive popup. Assistance may be required. Check Your Answer The Tigers win this game.
• Conclusion: Interactive popup. Assistance may be required. Check Your Answer The Tigers will play in the finals.
• Conditional Statement: Interactive popup. Assistance may be required. Check Your Answer If the Tigers win this game, then they will play in the finals.

To summarize:
The hypothesis is the If or p part of the sentence.
The conclusion is the Then or q part of the sentence.