Question

assume that the set of given information is true. use the law of detachment to form a conclusion about the situation. if a quadrilateral has one pair of opposite sides that are parallel and congruent, then the quadrilateral is a parallelogram. the quadrilateral has one pair of opposite sides that are parallel and congruent. choose the correct conclusion below. a. if a parallelogram has one pair of opposite sides that are parallel and congruent, then the parallelogram is a quadrilateral. b. the quadrilateral is a parallelogram. c. all quadrilaterals are parallelograms. d. all parallelograms are quadrilaterals. e. the parallelogram is a quadrilateral.

Explanation:

The Law of Detachment states that if we have a conditional statement \( p
ightarrow q \) (where \( p \) is the hypothesis and \( q \) is the conclusion) and we know that \( p \) is true, then we can conclude that \( q \) is true.

In this problem, the conditional statement is: "If a quadrilateral has one pair of opposite sides that are parallel and congruent, then the quadrilateral is a parallelogram." Here, \( p \) is "a quadrilateral has one pair of opposite sides that are parallel and congruent" and \( q \) is "the quadrilateral is a parallelogram."

We are given that "The quadrilateral has one pair of opposite sides that are parallel and congruent," which means \( p \) is true. By the Law of Detachment, we can conclude \( q \), which is "The quadrilateral is a parallelogram."

Let's analyze the other options:

• Option A: This is a misstatement of the original conditional and does not follow from the Law of Detachment.
• Option C: The original statement only applies to quadrilaterals with one pair of opposite sides parallel and congruent, not all quadrilaterals.
• Option D: The original statement is about quadrilaterals becoming parallelograms, not the reverse about all parallelograms being quadrilaterals (which is true by definition but not the conclusion here).
• Option E: This is also a misstatement and does not follow from the given information.

Answer:

B. The quadrilateral is a parallelogram.