investigating geometry online homework 2 1 conditional statements

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Conditional Statement If Then's Defined in Geometry - 15+ Examples!

// Last Updated: January 21, 2020 - Watch Video //

In today’s geometry lesson , you’re going to learn all about conditional statements!

Jenn, Founder Calcworkshop ® , 15+ Years Experience (Licensed & Certified Teacher)

We’re going to walk through several examples to ensure you know what you’re doing.

In addition, this lesson will prepare you for deductive reasoning and two column proofs later on.

Here we go!

What are Conditional Statements?

To better understand deductive reasoning, we must first learn about conditional statements.

A conditional statement has two parts: hypothesis ( if ) and conclusion ( then ).

In fact, conditional statements are nothing more than “If-Then” statements!

Sometimes a picture helps form our hypothesis or conclusion. Therefore, we sometimes use Venn Diagrams to visually represent our findings and aid us in creating conditional statements.

But to verify statements are correct, we take a deeper look at our if-then statements. This is why we form the converse , inverse , and contrapositive of our conditional statements.

What is the Converse of a Statement?

Well, the converse is when we switch or interchange our hypothesis and conclusion.

Conditional Statement : “If today is Wednesday, then yesterday was Tuesday.”

Hypothesis : “If today is Wednesday” so our conclusion must follow “Then yesterday was Tuesday.”

So the converse is found by rearranging the hypothesis and conclusion, as Math Planet accurately states.

Converse : “If yesterday was Tuesday, then today is Wednesday.”

What is the Inverse of a Statement?

Now the inverse of an If-Then statement is found by negating (making negative) both the hypothesis and conclusion of the conditional statement.

So using our current conditional statement, “If today is Wednesday, then yesterday was Tuesday”.

Inverse : “If today is not Wednesday, then yesterday was not Tuesday.”

What is a Contrapositive?

And the contrapositive is formed by interchanging the hypothesis and conclusion and then negating both.

Contrapositive : “If yesterday was not Tuesday, then today is not Wednesday”

What is a Biconditional Statement?

A statement written in “if and only if” form combines a reversible statement and its true converse. In other words the conditional statement and converse are both true.

Continuing with our initial condition, “If today is Wednesday, then yesterday was Tuesday.”

Biconditional : “Today is Wednesday if and only if yesterday was Tuesday.”

Examples of Conditional Statements

In the video below we will look at several harder examples of how to form a proper statement, converse, inverse, and contrapositive. And here’s a big hint…

Whenever you see “con” that means you switch! It’s like being a con-artist!

Moreover, we will detail the process for coming up with reasons for our conclusions using known postulates. We will review the ten postulates that we have learned so far, and add a few more problems dealing with perpendicular lines, planes, and perpendicular bisectors.

After this lesson, we will be ready to tackle deductive reasoning head-on, and feel confident as we march onward toward learning two-column proofs!

Conditional Statements – Lesson & Examples (Video)

• Introduction to conditional statements
• 00:00:25 – What are conditional statements, converses, and biconditional statements? (Examples #1-2)
• 00:05:21 – Understanding venn diagrams (Examples #3-4)
• 00:11:07 – Supply the missing venn diagram and conditional statement for each question (Examples #5-8)
• Exclusive Content for Member’s Only
• 00:17:48 – Write the statement and converse then determine if they are reversible (Examples #9-12)
• 00:29:17 – Understanding the inverse, contrapositive, and symbol notation
• 00:35:33 – Write the statement, converse, inverse, contrapositive, and biconditional statements for each question (Examples #13-14)
• 00:45:40 – Using geometry postulates to verify statements (Example #15)
• 00:53:23 – What are perpendicular lines, perpendicular planes and the perpendicular bisector?
• 00:56:26 – Using the figure, determine if the statement is true or false (Example #16)
• Practice Problems with Step-by-Step Solutions
• Chapter Tests with Video Solutions

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3.2.2: Conditional Statements

• Last updated
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• Page ID 74307

• Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia
• Rio Hondo College

3.2.2 Learning Objectives

• Determine whether a conditional statement is true or false

In logic a  statement is something that is either true or false. A statement like 3 < 5 is true; a statement like “a rat is a fish” is false. A statement like “\(x < 5\)” is true for some values of \(x\) and false for others. When an action is taken or not depending on the value of a statement, it forms a conditional .

Definition: Statement and Conditional

A statement is either true or false.

A conditional is a compound statement of the form

"if \(p\) then \(q\)" or "if \(p\) then \(q\), else \(s\)"

where \(p\) and \(q\) are both statements.

In common language, an example of a conditional statement would be “If it is raining, then we’ll go to the mall. Otherwise we’ll go for a hike.”

The statement “If it is raining” is the condition – this may be true or false for any given day. If the condition is true, then we will follow the first course of action, and go to the mall. If the condition is false, then we will use the alternative, and go for a hike.

"If \(p\), then \(q\)" can be stated different ways and still mean the same thing. The following statements all could be rewritten using the if, then format.  Here p represents "I receive my check tomorrow," and q represents "I pay off my debt."

\(\begin{array}{|l|l|} \hline \text { If p, then q} & \text { If I receive my check tomorrow, then I will pay off my debt. } \\ \hline \text { p is sufficient for q } & \text { Receiving my check tomorrow is sufficient for paying off my debt. } \\ \hline \text { p will lead to q } & \text { Receiving my check tomorrow will lead to the paying off my debt. }  \\ \hline \text { p implies q } & \text { Receiving my check tomorrow implies paying off my debt. }   \\ \hline \text { p is necessary for q } & \text { Receiving my check tomorrow is necessary for paying off my debt . } \\ \hline \text { q if p } & \text { I will pay off my debt if I receive my check tomorrow. }  \\ \hline \text { q whenever p } & \text { I will pay off my debt whenever I receive my check. .}  \\ \hline \end{array}\)

As we did earlier, we can create more complex conditions by using the operators and , or , and not to join simpler conditions together.

A parent might say to their child “if you clean your room and take out the garbage, then you can have ice cream.”

Here, there are two simpler conditions:

1) The child cleaning her room

2) The child taking out the garbage

Since these conditions were joined with and , the combined conditional will be true only if both simpler conditions are true; if either chore is not completed, then the parent’s condition is not met.

Notice that if the parent had said “if you clean your room or take out the garbage, then you can have ice cream”, then the child would need to complete only one chore to meet the condition.

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