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If a figure is a triangle, then it is a polygon.
1. Is this statement true or false?
2. What happens if we swap the hypothesis and conclusion?
"If a figure is a polygon, then it is a triangle."
3. Is this new statement true or false?
From one conditional statement, we can create three new related statements by swapping and/or negating the parts.
To write the converse, you swap the hypothesis and conclusion.
q → p
Original: If a figure is a square, then it is a rectangle. (True)
Converse: If a figure is a rectangle, then it is a square. (False)
To write the inverse, you negate both parts of the original statement.
~p → ~q
Original: If a figure is a square, then it is a rectangle. (True)
Inverse: If a figure is NOT a square, then it is NOT a rectangle. (False)
To write the contrapositive, you swap AND negate both parts.
~q → ~p
Original: If a figure is a square, then it is a rectangle. (True)
Contrapositive: If a figure is NOT a rectangle, then it is NOT a square. (True)
Statements that always have the same truth value are called logically equivalent.
The Original Statement is logically equivalent to the Contrapositive.
The Converse is logically equivalent to the Inverse.
When a conditional statement AND its converse are both TRUE, you can combine them.
We use the phrase "if and only if" (often written as "iff").
p ↔ q
A figure is a triangle if and only if it is a polygon with three sides.
Choose your level and solve the problems.
Given: "If it is snowing, then it is cold."
Given: "If a number is divisible by 10, then it is divisible by 5."
On your notecard, analyze the following statement:
"If two angles are vertical angles, then they are congruent."
1. Write the converse of this statement.
2. Write the contrapositive of this statement.