Question

because the slopes of $overline{np}$ and $\boxed{}$ are both $\boxed{}$, they are parallel.
because the slopes of $overline{pq}$ and $\boxed{}$ are both $\boxed{}$, they are parallel.
because both $\boxed{}$ are parallel, quadrilateral $npqr$ is a parallelogram by the $\boxed{}$.

options for blanks (from bottom to top, left to right): $-1$, $1$, $-\frac{4}{5}$, $\frac{4}{5}$, $-\frac{4}{3}$, $\frac{4}{3}$, $overline{nr}$, $overline{pr}$, $overline{qr}$, definition of a parallelogram, parallelogram opposite sides converse, parallelogram opposite angles converse, opposite sides parallel and congruent theorem, parallelogram diagonals converse, pairs of opposite sides, pairs of bisected diagonals

Explanation:

To solve this problem, we analyze the properties of a parallelogram (opposite sides are parallel) and use the given slope options and side labels.

First Statement (Slopes of \( \overline{NP} \) and \( \overline{QR} \))

For two lines to be parallel, their slopes must be equal. From the options, we identify the pair of sides (\( \overline{NP} \) and \( \overline{QR} \)) and their common slope. The slope here is \( \boldsymbol{\frac{4}{5}} \) (matching the slope of \( \overline{NP} \) and \( \overline{QR} \)).

Second Statement (Slopes of \( \overline{PQ} \) and \( \overline{NR} \))

Similarly, \( \overline{PQ} \) and \( \overline{NR} \) are opposite sides of the quadrilateral. Their common slope is \( \boldsymbol{-\frac{4}{3}} \) (ensuring they are parallel).

Third Statement (Parallel Sides and Parallelogram Definition)

A parallelogram is defined as a quadrilateral with pairs of opposite sides parallel. Thus, since \( \overline{NP} \parallel \overline{QR} \) and \( \overline{PQ} \parallel \overline{NR} \) (two pairs of opposite sides), quadrilateral \( NPQR \) is a parallelogram by the definition of a parallelogram (or "Parallelogram Opposite Sides Converse," which states that if both pairs of opposite sides of a quadrilateral are parallel, it is a parallelogram).

Final Answers (Filling in the Blanks):

1. Because the slopes of \( \overline{NP} \) and \( \boldsymbol{\overline{QR}} \) are both \( \boldsymbol{\frac{4}{5}} \), they are parallel.
2. Because the slopes of \( \overline{PQ} \) and \( \boldsymbol{\overline{NR}} \) are both \( \boldsymbol{-\frac{4}{3}} \), they are parallel.
3. Because both \( \boldsymbol{\text{pairs of opposite sides}} \) ( \( \overline{NP} \parallel \overline{QR} \) and \( \overline{PQ} \parallel \overline{NR} \) ) are parallel, quadrilateral \( NPQR \) is a parallelogram by the \( \boldsymbol{\text{definition of a parallelogram}} \) (or "Parallelogram Opposite Sides Converse").

(Note: The exact labels depend on the problem’s context, but the key is identifying opposite sides with equal slopes and applying the parallelogram definition.)

Answer:

To solve this problem, we analyze the properties of a parallelogram (opposite sides are parallel) and use the given slope options and side labels.

First Statement (Slopes of \( \overline{NP} \) and \( \overline{QR} \))

For two lines to be parallel, their slopes must be equal. From the options, we identify the pair of sides (\( \overline{NP} \) and \( \overline{QR} \)) and their common slope. The slope here is \( \boldsymbol{\frac{4}{5}} \) (matching the slope of \( \overline{NP} \) and \( \overline{QR} \)).

Second Statement (Slopes of \( \overline{PQ} \) and \( \overline{NR} \))

Similarly, \( \overline{PQ} \) and \( \overline{NR} \) are opposite sides of the quadrilateral. Their common slope is \( \boldsymbol{-\frac{4}{3}} \) (ensuring they are parallel).

Third Statement (Parallel Sides and Parallelogram Definition)

A parallelogram is defined as a quadrilateral with pairs of opposite sides parallel. Thus, since \( \overline{NP} \parallel \overline{QR} \) and \( \overline{PQ} \parallel \overline{NR} \) (two pairs of opposite sides), quadrilateral \( NPQR \) is a parallelogram by the definition of a parallelogram (or "Parallelogram Opposite Sides Converse," which states that if both pairs of opposite sides of a quadrilateral are parallel, it is a parallelogram).

Final Answers (Filling in the Blanks):

1. Because the slopes of \( \overline{NP} \) and \( \boldsymbol{\overline{QR}} \) are both \( \boldsymbol{\frac{4}{5}} \), they are parallel.
2. Because the slopes of \( \overline{PQ} \) and \( \boldsymbol{\overline{NR}} \) are both \( \boldsymbol{-\frac{4}{3}} \), they are parallel.
3. Because both \( \boldsymbol{\text{pairs of opposite sides}} \) ( \( \overline{NP} \parallel \overline{QR} \) and \( \overline{PQ} \parallel \overline{NR} \) ) are parallel, quadrilateral \( NPQR \) is a parallelogram by the \( \boldsymbol{\text{definition of a parallelogram}} \) (or "Parallelogram Opposite Sides Converse").

(Note: The exact labels depend on the problem’s context, but the key is identifying opposite sides with equal slopes and applying the parallelogram definition.)