# 2.11: If Then Statements

• • Contributed by CK12
• CK12

Hypothesis followed by a conclusion in a conditional statement.

## Conditional Statements

A conditional statement (also called an if-then statement) is a statement with a hypothesis followed by a conclusion. 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. Figure $$\PageIndex{1}$$

If-then statements might not always be written in the “if-then” form. Here are some examples of conditional statements:

• 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 $$x$$, then $$x$$ is an even number.
• Statement 4: I’ll be a millionaire when I win the lottery.
• 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.” Statement 2 has the hypothesis after the conclusion. If the word “if” is in the middle of the statement, then the hypothesis is after it. The statement can be rewritten: If the weather is nice, then I will wash the car. Statement 4 uses the word “when” instead of “if” and is like Statement 2. It can be written: If I win the lottery, then I will be a millionaire. Statement 5 “if” and “then” are not there. It can be rewritten: If a triangle is equiangular, then it is equilateral.

What if you were given a statement like "All squares are rectangles"? How could you determine the hypothesis and conclusion of this statement?

Example $$\PageIndex{1}$$

Determine the hypothesis and conclusion: I'll bring an umbrella if it rains.

Solution

Hypothesis: "It rains." Conclusion: "I'll bring an umbrella."

Example $$\PageIndex{2}$$

Determine the hypothesis and conclusion: All right angles are $$90^{\circ}$$.

Solution

Hypothesis: "An angle is right." Conclusion: "It is $$90^{\circ}$$."

Example $$\PageIndex{3}$$

Use the statement: I will graduate when I pass Calculus.

Rewrite in if-then form and determine the hypothesis and conclusion.

Solution

This statement can be rewritten as If I pass Calculus, then I will graduate. The hypothesis is “I pass Calculus,” and the conclusion is “I will graduate.”

Example $$\PageIndex{4}$$

Use the statement: All prime numbers are odd.

Rewrite in if-then form, determine the hypothesis and conclusion, and determine whether this is a true statement.

Solution

This statement can be rewritten as If a number is prime, then it is odd. The hypothesis is "a number is prime" and the conclusion is "it is odd". This is not a true statement (remember that not all conditional statements will be true!) since 2 is a prime number but it is not odd.

Example $$\PageIndex{5}$$

Determine the hypothesis and conclusion: Sarah will go to the store if Riley does the laundry.

Solution

The statement can be rewritten as "If Riley does the laundry then Sarah will go to the store." The hypothesis is "Riley does the laundry" and the conclusion is "Sarah will go to the store."

## Review

Determine the hypothesis and the conclusion for each statement.

1. If 5 divides evenly into $$x$$, then $$x$$ 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 $$x=3$$, then $$x^2=9$$.
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.

## Vocabulary

Term Definition
Conditional Statement A conditional statement (or 'if-then' statement) is a statement with a hypothesis followed by a conclusion.
Angle A geometric figure formed by two rays that connect at a single point or vertex.
antecedent The antecedent is the first, or “if,” part of a conditional statement.
apodosis The “then” part of an if-then statement is called the conclusion, consequent or apodosis.
conclusion The conclusion of a conditional statement is the result of the hypothesis.
consequent The “then” part of an if-then statement is called the conclusion, consequent or apodosis.
hypothesis The hypothesis is the first, or “if,” part of a conditional statement.
if-then statement An if-then statement is another name for a conditional statement.
protasis The protasis is the first, or “if,” part of a conditional statement.
set A set is a collection of numbers, letters or anything.
set theory Set theory studies the relationships of sets and subsets.
Subset A subset is a collection of numbers or objects within a larger set.