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CAHSEE Math Exam: Help and Review22 chapters | 255 lessons | 13 flashcard sets

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Lesson Transcript

Instructor:
*Beverly Maitland-Frett*

Beverly has taught mathematics at the high school level and has a doctorate in teaching and learning.

In this lesson, you'll learn how to define and recognize a biconditional statement. We'll review conditional statements and their converses and look at examples of each. We'll also discuss the criteria for writing biconditional statements.

When you were a child, your parents might have said, 'If you are good, then I'll give you a surprise.' This is an example of a conditional statement. Biconditional statements are partially formed from conditional statements. But before we can fully explore biconditional statements, we have to understand conditional statements and their converse statements.

**Conditional statements** use the words 'if' and 'then.' They have two parts: a **hypothesis** and a **conclusion**. Our example had two parts: (1) If you are good, (2) then I'll give you a surprise.

- Our hypothesis: you are good
- Our conclusion: I'll give you a surprise

Let's try another one: If it is sunny, then it will be a hot day.

- Our hypothesis: it is sunny
- Our conclusion: it will be a hot day

The thing about conditional statement is that they are not necessarily true. Our last example about the sun is not true because the sun shines in the winter, too. Therefore, as long as there is an 'if' and a hypothesis, along with a 'then' and a conclusion, you have a conditional statement.

Remember how we said that conditional statements have a hypothesis and a conclusion? Well, in **converse statements**, the hypothesis and the conclusion exchange places. When we revisit our example about the sun, the converse statement would read: If it is a hot day, then it is sunny.

- Our hypothesis: it is a hot day
- Our conclusion: it is sunny

Let's look at a new example in relationship to geometry:

- Conditional statement: If a polygon is a triangle, then it has three sides.
- Converse statement: If a polygon has three sides, then it is a triangle.

Notice that both of these statements are true. Now, let's move on to biconditional statements.

So, now that we've reviewed conditional statements and their converses, let's take a look at biconditional statements. **Biconditional statements** do not use the key words 'if' and 'then.' Biconditional statements are true statements that combine the hypothesis and the conclusion with the key words 'if and only if.'

For example, the statement will take this form: (hypothesis) if and only if (conclusion). We could also write it this way: (conclusion) if and only if (hypothesis). If you figured out that both the conditional and converse statements have to be true for a biconditional statement to exist in geometry, you are correct. It's like a reversible jacket; you can wear it on both sides.

Let's rewrite our last example:

- Conditional statement: If a polygon has three sides, then it is a triangle.
- Converse statement: If a polygon is a triangle, then it has three sides.

Since both are statements are true, we can go ahead and make our biconditional statements:

- A polygon is a triangle 'if and only if' it has three sides.
- A polygon has three sides 'if and only if' it is a triangle.

Since we can write two biconditional statements, we could also define them as compound statements, since both the conditional and the converse statements have to be true. Let's practice some more.

It may make more sense to you if we write our statements in groups, so let's do that. We'll start with some examples from everyday life and some that relate to geometry.

**Conditional:** If I finish all the requirements, then I will graduate from high school.**Converse:** If I graduated from high school, then I finished all the requirements.**Biconditional:** I will graduate from high school if and only if I finish all the requirements. (or) I finished the requirements if and only if I graduated from high school.

**Conditional:** If a child is an infant, then he is between 0 and 12 months of age.**Converse:** If a child is between 0 and 12 months of age, then the child is an infant.**Biconditional:** A child is an infant if and only if he is between 0 and 12 months of age. (or) A child is between 0 and 12 months of age if and only if he is an infant.

**Conditional:** If a shape has four congruent sides, then it is a parallelogram.**Converse:** If a shape is a parallelogram, then it has four congruent sides.**Biconditional:** Sorry we can't write one, because the converse statement is not true. Not all parallelograms have four congruent sides, as in the case of a rectangle.

**Conditional:** If an angle measures 90 degrees, then it is a right angle.**Converse:** If an angle is a right angle, then it measures 90 degrees.**Biconditional:** An angle measures 90 degrees if and only if it is a right angle. (or) An angle is a right angle if and only if it measures 90 degrees.

So, let's put all of this together.

**Conditional statements**are 'if' and 'then' statements that use both a hypothesis and a conclusion. They take this form: If (some hypothesis) then (some conclusion).**Converse statements**are statements in which the hypothesis and the conclusion exchange places.- If we want to write a biconditional statement, both statements have to be true.
**Biconditional statements**use the phrase 'if and only if' to join the hypothesis and the conclusion.- Never begin a biconditional statement with 'if and only if.' Many people confuse this with the conditional. Therefore, it is incorrect to say: If and only if it is a triangle, then it has three sides.

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CAHSEE Math Exam: Help and Review22 chapters | 255 lessons | 13 flashcard sets

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