Blog Entry #2

Hello, I am back. This is my 2nd blog entry. So our geometry teacher gave us an assignment involving biconditional statements and its conditional and converse on lesson 2-4 on our geometry textbooks. We were given three questions. In our first question, we had to write the definition of a biconditional statement as a biconditional statement. Next, we had to use the conditional and converse within the statement to explain why our biconditional is true. 

Definition of biconditional: A statement that can be written in the form "p if and only if q." 

Biconditional: A statement is a biconditional if and only if it is written in the form "p if and only if q." 

Conditional: If a statement is written in the form "p if and only if q," then it is a biconditional. (Truth Value: True)

Converse: If a statement is written in the form "p if and only if q," then it is a biconditional. (Truth Value: True)

In order for a biconditional to be true, both its conditional and converse must be true. Both the conditional and converse of the biconditional are true; therefore, the biconditional is true. 



In our second question, we had to use the definition of an angle bisector to explain what is meant by the statement "A good definition is reversible."

Definition of Angle Bisector: A ray that divides an angle into two congruent angles.

(The red line is the angle bisector)

The statement "a good definition is reversible" applies to the definition of "angle bisector." The definition of "angle bisector" can be written as a biconditional, conditional, and converse.

Biconditional: A ray is an angle bisector if and only if it divides an angle into two congruent angles. (Truth Value: True)

Conditional: If a ray is an angle bisector, then it divides an angle into two congruent angles. (Truth Value: True)

Converse: If a ray divides an angle into two congruent angles, then it is an angle bisector. (Truth Value: True)

All statements are true; In each statement, the form changes but still keeps its truth value. The definition of "angle bisector" is a "good definition" because it is "reversible."

And finally, our third question asks us to write the two conditional statements that make up the biconditional "You will get a traffic ticket if and only if you are speeding." Then we have to determine whether the biconditional true or false and explain our answer. 

Conditional: If you get a traffic ticket, then you are speeding. (Truth Value: False, because speeding isn't the only traffic violation you can do to get a traffic ticket.)

Converse: If you are speeding, then you get a traffic ticket, (Truth Value: False, because drivers only get traffic tickets when they are caught speeding. Not all drivers who speed get traffic tickets.)

The biconditional is false because both conditional and converse were false. In order for the biconditional to be true, the conditional and converse must both be true. But since both are false, then the biconditional is false.