Quadrilateral ABCD plotted on a coordinate grid with labeled vertices.

Problem Analysis:

The image shows a quadrilateral \(ABCD\) with the following coordinates:
- \(A = (1, 4)\)
- \(B = (0, 1)\)
- \(C = (3, 1)\)
- \(D = (4, 4)\)

The task is to determine the type of quadrilateral \(ABCD\) and verify its properties. Let's solve this step by step.

---

Step 1: Calculate the lengths of the sides

To identify the type of quadrilateral, we first calculate the lengths of the sides \(AB\), \(BC\), \(CD\), and \(DA\). The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\]

#### Length of \(AB\):
\[
AB = \sqrt{(1 - 0)^2 + (4 - 1)^2} = \sqrt{1^2 + 3^2} = \sqrt{1 + 9} = \sqrt{10}
\]

#### Length of \(BC\):
\[
BC = \sqrt{(3 - 0)^2 + (1 - 1)^2} = \sqrt{3^2 + 0^2} = \sqrt{9 + 0} = 3
\]

#### Length of \(CD\):
\[
CD = \sqrt{(4 - 3)^2 + (4 - 1)^2} = \sqrt{1^2 + 3^2} = \sqrt{1 + 9} = \sqrt{10}
\]

#### Length of \(DA\):
\[
DA = \sqrt{(4 - 1)^2 + (4 - 4)^2} = \sqrt{3^2 + 0^2} = \sqrt{9 + 0} = 3
\]

So, the side lengths are:
\[
AB = \sqrt{10}, \quad BC = 3, \quad CD = \sqrt{10}, \quad DA = 3
\]

---

Step 2: Check for parallel sides

Next, we calculate the slopes of the sides to determine if any pairs of sides are parallel. The slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

#### Slope of \(AB\):
\[
m_{AB} = \frac{4 - 1}{1 - 0} = \frac{3}{1} = 3
\]

#### Slope of \(BC\):
\[
m_{BC} = \frac{1 - 1}{3 - 0} = \frac{0}{3} = 0
\]

#### Slope of \(CD\):
\[
m_{CD} = \frac{4 - 1}{4 - 3} = \frac{3}{1} = 3
\]

#### Slope of \(DA\):
\[
m_{DA} = \frac{4 - 4}{1 - 4} = \frac{0}{-3} = 0
\]

From the slopes:
- \(AB\) and \(CD\) have the same slope (\(3\)), so \(AB \parallel CD\).
- \(BC\) and \(DA\) have the same slope (\(0\)), so \(BC \parallel DA\).

Since both pairs of opposite sides are parallel, \(ABCD\) is a parallelogram.

---

Step 3: Check for right angles

To determine if \(ABCD\) is a rectangle, we need to check if any of the angles are right angles. We can do this by checking if the product of the slopes of adjacent sides is \(-1\) (indicating perpendicularity).

#### Check \(AB\) and \(BC\):
\[
m_{AB} \cdot m_{BC} = 3 \cdot 0 = 0 \quad (\text{not } -1)
\]

#### Check \(BC\) and \(CD\):
\[
m_{BC} \cdot m_{CD} = 0 \cdot 3 = 0 \quad (\text{not } -1)
\]

#### Check \(CD\) and \(DA\):
\[
m_{CD} \cdot m_{DA} = 3 \cdot 0 = 0 \quad (\text{not } -1)
\]

#### Check \(DA\) and \(AB\):
\[
m_{DA} \cdot m_{AB} = 0 \cdot 3 = 0 \quad (\text{not } -1)
\]

Since none of the adjacent sides are perpendicular, \(ABCD\) is not a rectangle.

---

Step 4: Check for equal diagonals

To determine if \(ABCD\) is a rhombus, we calculate the lengths of the diagonals \(AC\) and \(BD\).

#### Length of diagonal \(AC\):
\[
AC = \sqrt{(3 - 1)^2 + (1 - 4)^2} = \sqrt{2^2 + (-3)^2} = \sqrt{4 + 9} = \sqrt{13}
\]

#### Length of diagonal \(BD\):
\[
BD = \sqrt{(4 - 0)^2 + (4 - 1)^2} = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5
\]

Since the diagonals are not equal, \(ABCD\) is not a rhombus.

---

Step 5: Conclusion

From the above steps:
1. Opposite sides are parallel, so \(ABCD\) is a parallelogram.
2. Adjacent sides are not perpendicular, so it is not a rectangle.
3. Diagonals are not equal, so it is not a rhombus.

Thus, the quadrilateral \(ABCD\) is a parallelogram.

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Final Answer:

\[
\boxed{\text{parallelogram}}
\]