What if you were given a statement like "All squares are rectangles"? How could you determine the hypothesis and conclusion of this statement? After completing this Concept, you'll be able to rewrite statements in if-then, or conditional, form.
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James Sousa: If-Then Statements and Converses
Guidance
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.
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 , then 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.
Example A
Use the statement: I will graduate when I pass Calculus.
a) Rewrite in if-then form.
b) Determine the hypothesis and conclusion.
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 B
Use the statement: All prime numbers are odd.
a) Rewrite in if-then form.
b) Determine the hypothesis and conclusion.
c) Is this a true statement?
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 C
Determine the hypothesis and conclusion: Sarah will go to the store if Riley does the laundry.
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."
Guided Practice
Determine the hypothesis and conclusion:
1. I'll bring an umbrella if it rains.
2. If I win the game, then I will get a prize.
3. All right angles are .
Answers:
1. Hypothesis: "It rains." Conclusion: "I'll bring an umbrella."
2. Hypothesis: "I win the game." Conclusion: "I will get a prize."
3. Hypothesis: "An angle is right." Conclusion: "It is ."
Practice
Determine the hypothesis and the conclusion for each statement.
- If 5 divides evenly into , then 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 , then .
- 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.