Answer :
Let's examine each statement to determine their truth value and whether they convey the same meaning.
Statement 1: If it is a triangle, then it has three sides.
This statement is true by definition. A triangle is a polygon with three sides. Therefore, if a shape is a triangle, it inherently has three sides.
Statement 2: If it does not have three sides, then it is not a triangle.
This statement reflects the contrapositive of the first statement. In logical terms, a contrapositive of a statement "If A, then B" is "If not B, then not A." The contrapositive of a true statement is always true. Therefore, since the first statement is true, its contrapositive is also true.
Now, we need to determine whether these statements have the same meaning. The first statement asserts a direct mathematical definition, implying that three sides are definitive of a triangle. The second statement, being the contrapositive, essentially reinforces the same condition from a different perspective. Logically, both statements are saying the same thing about what it means to be a triangle (having three sides).
A contrapositive of a statement is logically equivalent to the original statement. Thus, these statements are logically equivalent and have the same meaning.
Conclusion:
- Both statements are true.
- Both statements have the same meaning.
Therefore, the correct answer is:
Both statements are true and they have the same meaning.
Statement 1: If it is a triangle, then it has three sides.
This statement is true by definition. A triangle is a polygon with three sides. Therefore, if a shape is a triangle, it inherently has three sides.
Statement 2: If it does not have three sides, then it is not a triangle.
This statement reflects the contrapositive of the first statement. In logical terms, a contrapositive of a statement "If A, then B" is "If not B, then not A." The contrapositive of a true statement is always true. Therefore, since the first statement is true, its contrapositive is also true.
Now, we need to determine whether these statements have the same meaning. The first statement asserts a direct mathematical definition, implying that three sides are definitive of a triangle. The second statement, being the contrapositive, essentially reinforces the same condition from a different perspective. Logically, both statements are saying the same thing about what it means to be a triangle (having three sides).
A contrapositive of a statement is logically equivalent to the original statement. Thus, these statements are logically equivalent and have the same meaning.
Conclusion:
- Both statements are true.
- Both statements have the same meaning.
Therefore, the correct answer is:
Both statements are true and they have the same meaning.
