To approach this question, we need to understand and analyze the different types of logical statements based on the given true statement. The given statement is:
True Statement:
"If two lines are not perpendicular, then they do not intersect to form right angles."
Let's break this down step-by-step:
1. Contrapositive:
The contrapositive of a statement is formed by negating both the hypothesis and the conclusion, and then swapping them. For the given statement, the contrapositive would be:
"If two lines do not intersect to form right angles, then they are not perpendicular."
The contrapositive of a true statement is always true. Thus, this statement is true.
2. Inverse:
The inverse of a statement is formed by negating both the hypothesis and the conclusion. For the given statement, the inverse would be:
"If two lines are perpendicular, then they intersect to form right angles."
By analyzing this statement, we note that if two lines are indeed perpendicular, they must intersect to form right angles. Hence, this statement is logically consistent and true.
3. Converse:
The converse of a statement is formed by swapping the hypothesis and the conclusion. For the given statement, the converse would be:
"If two lines do not intersect to form right angles, then they are not perpendicular."
Similarly, since perpendicular lines necessarily form right angles upon intersection, any pair of lines that do not form right angles upon intersection cannot be perpendicular. This statement, therefore, is logically consistent and true.
In conclusion, given the true statement:
"If two lines are not perpendicular, then they do not intersect to form right angles."
We determined that:
- The contrapositive: "If two lines do not intersect to form right angles, then they are not perpendicular," is true.
- The inverse: "If two lines are perpendicular, then they intersect to form right angles," is true.
- The converse: "If two lines do not intersect to form right angles, then they are not perpendicular," is true.
Therefore, all these related statements —the contrapositive, inverse, and converse— are true.
