Which best explains if quadrilateral WXYZ can be a parallelogram?
WXYZ can be a parallelogram with one pair of sides measuring 15 mm and the other pair measuring 9 mm.
WXYZ can be a parallelogram with one pair of sides measuring 15 mm and the other pair measuring 7 mm.
WXYZ cannot be a parallelogram because there are three different values for $x$ when each expression is set equal to 15.
WXYZ cannot be a parallelogram because the value of $x$ that makes one pair of sides congruent does not make the other pair of sides congruent.
Answer (1)
A parallelogram requires opposite sides to be congruent.
Options 1 and 2 describe parallelograms with given side lengths, which are possible.
Option 3 suggests a contradiction in side length uniqueness, implying it cannot be a parallelogram.
Option 4 directly addresses the congruence of opposite sides, providing the best explanation.
WXYZ cannot be a parallelogram because the value of x that makes one pair of sides congruent does not make the other pair of sides congruent.
Explanation
Understanding Parallelograms Let's analyze the properties of a parallelogram to determine which statement best explains when quadrilateral WXYZ can be a parallelogram. A key property of a parallelogram is that its opposite sides must be congruent (equal in length).
Analyzing Each Option
Option 1: WXYZ can be a parallelogram with one pair of sides measuring 15 mm and the other pair measuring 9 mm. This is possible if the opposite sides are equal. So, two sides are 15 mm and the other two are 9 mm. This aligns with the properties of a parallelogram.
Option 2: WXYZ can be a parallelogram with one pair of sides measuring 15 mm and the other pair measuring 7 mm. Similar to option 1, this is possible if the opposite sides are equal. Two sides are 15 mm and the other two are 7 mm. This also aligns with the properties of a parallelogram.
Option 3: WXYZ cannot be a parallelogram because there are three different values for x when each expression is set equal to 15. This statement implies that the side lengths are defined by expressions involving x . If there are three different values of x that result in a side length of 15, this contradicts the uniqueness required for side lengths in a parallelogram. This suggests the quadrilateral cannot be a parallelogram.
Option 4: WXYZ cannot be a parallelogram because the value of x that makes one pair of sides congruent does not make the other pair of sides congruent. This statement also implies that side lengths are defined by expressions involving x . If the value of x that makes one pair of sides congruent does not make the other pair congruent, then the quadrilateral cannot be a parallelogram. This directly relates to the definition of a parallelogram.
Choosing the Best Explanation Comparing options 3 and 4, option 4 provides a more direct and general explanation based on the fundamental property of parallelograms (opposite sides congruent). Option 3 is more specific and depends on the context of 'three different values for x'. Therefore, option 4 is the better explanation.
Examples
In architecture, understanding parallelograms is crucial for designing stable and aesthetically pleasing structures. For example, when designing a bridge, engineers use parallelogram structures to distribute weight evenly. If the opposite sides of the parallelogram are not congruent, the structure may be unstable and prone to collapse. Therefore, ensuring the congruence of opposite sides is essential for structural integrity.
