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Nico and Karina wrote these statements in their math journals.

\begin{tabular}{|l|l|}
\hline \multicolumn{1}{|c|}{Nico's Statement} & \multicolumn{1}{c|}{Karina's Statement} \\
\hline \begin{tabular}{l}
A 4-sided polygon with at least one pair of parallel sides \\
must be a parallelogram.
\end{tabular} & A 4-sided polygon with at least one pair of parallel sides \\
must be a trapezoid.
\end{tabular}

Which describes the accuracy of their statements and why?

A. Nico is correct because if one pair of sides in a quadrilateral is parallel, the other pair must be also.

B. Karina is correct because a trapezoid has exactly one pair of parallel sides.

C. Both are correct because a 4-sided polygon with at least one pair of parallel sides must be both a parallelogram and a trapezoid.

D. Neither is correct because a 4-sided polygon with at least one pair of parallel sides could be either a parallelogram or a trapezoid.



Answer :

GinnyAnswer
Let's break down the problem and examine the statements made by Nico and Karina regarding 4-sided polygons, also known as quadrilaterals.

### Definitions:
1. Parallelogram: A quadrilateral with two pairs of parallel sides.
2. Trapezoid: A quadrilateral with exactly one pair of parallel sides. (Note that some definitions of trapezoid might include cases where there is at least one pair of parallel sides, but for the purposes of this problem, we'll use the definition specifying exactly one pair.)

### Statements:
Nico's Statement:
"A 4-sided polygon with at least one pair of parallel sides must be a parallelogram."

Karina's Statement:
"A 4-sided polygon with at least one pair of parallel sides must be a trapezoid."

### Analysis:
1. Nico's Statement:
- A parallelogram is specifically defined as having two pairs of parallel sides.
- If a quadrilateral has at least one pair of parallel sides, this does not necessarily mean that the other pair is also parallel.
- Therefore, Nico’s statement is incorrect because a quadrilateral with at least one pair of parallel sides does not have to be a parallelogram; it could simply be a trapezoid.

2. Karina’s Statement:
- A trapezoid is defined as having exactly one pair of parallel sides.
- If a quadrilateral has at least one pair of parallel sides, it fits the definition of a trapezoid (assuming the definition requiring exactly one pair of parallel sides).
- Therefore, Karina's statement is correct because she states that a 4-sided polygon with at least one pair of parallel sides is a trapezoid, which fits the given definition.

### Conclusion:
- Nico is not correct because a quadrilateral with at least one pair of parallel sides is not necessarily a parallelogram (it could be a trapezoid).
- Karina is correct because a quadrilateral with at least one pair of parallel sides fits the definition of a trapezoid.

Thus, the accuracy of their statements is:
Karina is correct because a trapezoid has exactly one pair of parallel sides.

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