9/12/13- Biconditionals and Converse Statements

Think about the definition of congruent angles:  Congruent angles have equal measures.  Now, rewrite that statement as an “If… then…” statement:

1.) If angles are congruent, then they have the same measure.

or

2.) If angles have the same measure, then they are congruent.

The definition can be rewritten in two ways, and they are each completely true and valid!  Notice how the two statements are similar, but also different.  They both deal with the ideas of congruence and equal measures, but they have different conditions and conclusions.  The first uses the condition that the angles are congruent, and arrives at the conclusion that the angles must have the same measure, while statement 2 uses the condition that the angles have the same measure to arrive at the conclusion that they must be congruent.  These two different statements are used differently.  Sometimes your given info about congruence, and use that to talk about equal measures, while other times you know the measures are equal, and so you can say angles are congruent.  It’s all about the order in which things come about.

When you switch the condition and conclusion of a statement to get a news statement, that new statement is called the Converse of the original statement (Think Con as in contrary, meaning against, which could also be thought of as switched, and then verse, meaning a statement, for example, a line from a song.  Combining those ideas gives us Converse- switched statement).

This idea of switching the condition and conclusion of a statement to get a new, valid statement is actually kind of rare though.  Not every statement has this property.  Definitions always do, as a word implies the statement that defines it, and a definition implies the word it defines (the idea of working “both ways”).  Statements that have this convenient property are called Biconditionals.

Biconditional– a true or valid statement that remains true or valid even after switching its condition and conclusion. (The original statement and its converse are both true).

Ex.) If it is Thursday, then school starts at 8:25. (True statement)

Converse: If school starts at 8:25, then it is Thursday.  (True statement)

This works either way, so the statement is considered to be a biconditional.

Non-ex.)  If I am in class at 7:25, then it is first period.

Converse: If it is first period, then I am in class at 7:25.  (not true, since on Thursday, when you are in first period, it’s 8:25, not 7:25)

Here, the converse is a false statement, so the statement is not a biconditional.

False Statements:  It only takes one example of a statement not being true to prove a statement false.  Such an example is called a counter-example.

Now, we remind ourselves of the Corresponding Angles Postulate, but making sure to write it in “If…then…” form:

If two lines are parallel, then the corresponding angles are congruent.

Remember, the condition here is that the two lines are parallel, and the conclusion is that the corresponding angles must be congruent.  What does this statement look like if we switch the conclusion and condition?  Well, first of all, we would call this new statement, the Converse of the Corresponding Angles Postulate:

If the corresponding angles are congruent, then the lines are parallel.

Here, the condition is that the corresponding angles are congruent.  If this condition is met, we say that the lines must be parallel.  Note that this is not a theorem!  We haven’t proven anything!  The best we can do is agree that we can’t imagine the world being otherwise.  These kinds of unproven statements are called postulates.

The corresponding angles postulate was the basis for using given info about parallel lines to arrive at lots of convenient conclusions about angle pairs.

The converse of the corresponding angles postulate, is going to be the basis for using given facts about angles to arrive at the conclusion that the lines must be parallel.  This is a different direction.  The proofs will be very similar, but still slightly different.

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