A quadrilateral has vertices $$ A(4,5), B(2,4), C(4,3) $$, and $$ D(6,4) $$. Which statement about the quadrilateral is true? A. $$ A B C D $$ is a parallelogram with noncongruent adjacent sides. B. $$ A B C D $$ is a trapezoid with only one pair of parallel sides. C. $$ A B C D $$ is a rectangle with noncongruent adjacent sides. D. $$ A B C D $$ is a square. E. $$ A B C D $$ is a rhombus with non-perpendicular adjacent sides.

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We start by computing the lengths of all four sides.

1. Compute $$ AB $$ between $$ A(4,5) $$ and $$ B(2,4) $$:
   $$
   AB=\sqrt{(2-4)^2+(4-5)^2}=\sqrt{(-2)^2+(-1)^2}=\sqrt{4+1}=\sqrt{5}.
   $$

2. Compute $$ BC $$ between $$ B(2,4) $$ and $$ C(4,3) $$:
   $$
   BC=\sqrt{(4-2)^2+(3-4)^2}=\sqrt{(2)^2+(-1)^2}=\sqrt{4+1}=\sqrt{5}.
   $$

3. Compute $$ CD $$ between $$ C(4,3) $$ and $$ D(6,4) $$:
   $$
   CD=\sqrt{(6-4)^2+(4-3)^2}=\sqrt{(2)^2+(1)^2}=\sqrt{4+1}=\sqrt{5}.
   $$

4. Compute $$ DA $$ between $$ D(6,4) $$ and $$ A(4,5) $$:
   $$
   DA=\sqrt{(4-6)^2+(5-4)^2}=\sqrt{(-2)^2+(1)^2}=\sqrt{4+1}=\sqrt{5}.
   $$

Since all four sides are of equal length, the quadrilateral is equilateral.

Next, we check the slopes of opposite sides to determine if the quadrilateral is a parallelogram.

- Slope of $$ AB $$ (between $$ A(4,5) $$ and $$ B(2,4) $$):
  $$
  m_{AB}=\frac{4-5}{2-4}=\frac{-1}{-2}=\frac{1}{2}.
  $$
  
- Slope of $$ CD $$ (between $$ C(4,3) $$ and $$ D(6,4) $$):
  $$
  m_{CD}=\frac{4-3}{6-4}=\frac{1}{2}.
  $$

Thus, $$ AB \parallel CD $$.

- Slope of $$ BC $$ (between $$ B(2,4) $$ and $$ C(4,3) $$):
  $$
  m_{BC}=\frac{3-4}{4-2}=\frac{-1}{2}=-\frac{1}{2}.
  $$
  
- Slope of $$ DA $$ (between $$ D(6,4) $$ and $$ A(4,5) $$):
  $$
  m_{DA}=\frac{5-4}{4-6}=\frac{1}{-2}=-\frac{1}{2}.
  $$

Thus, $$ BC \parallel DA $$.

Because both pairs of opposite sides are parallel, the quadrilateral is a parallelogram. An equilateral parallelogram is a rhombus.

Finally, we check whether adjacent sides are perpendicular (which is required for a square and rectangle).

Consider the dot product of vectors $$ \overrightarrow{AB} $$ and $$ \overrightarrow{BC} $$. The vector $$ \overrightarrow{AB}=(2-4,\,4-5)=(-2,-1) $$ and $$ \overrightarrow{BC}=(4-2,\,3-4)=(2,-1) $$.

The dot product is:
$$
\overrightarrow{AB}\cdot\overrightarrow{BC}=(-2)(2)+(-1)(-1)=-4+1=-3.
$$
Since $$-3\neq 0$$, the sides are not perpendicular.

Thus, $$ ABCD $$ is a rhombus with non-perpendicular adjacent sides. This matches choice E.
$$ ABCD $$ is a rhombus with non-perpendicular adjacent sides.

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