Let’s solve this step-by-step:

1. Plot the Points on a Coordinate Plane

The points given are:

• $A(-2, 2)$
• $B(3, 2)$
• $C(-2, -4)$
• $D(3, -4)$

We can place these points in a coordinate plane. The x-values and y-values suggest these points will form a rectangle since the x-coordinates of $A$ and $C$ are the same, and the x-coordinates of $B$ and $D$ are the same, indicating vertical alignment. Similarly, the y-coordinates of $A$ and $B$ are the same, as are $C$ and $D$, indicating horizontal alignment.

---

2. Check Congruent Segments

Step 1: Length of $\overline{AB}$

Use the distance formula:
$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
For $\overline{AB}$, where $A(-2, 2)$ and $B(3, 2)$:
$d_{AB} = \sqrt{(3 - (-2))^2 + (2 - 2)^2} = \sqrt{(3 + 2)^2 + 0} = \sqrt{25} = 5$

Step 2: Length of $\overline{CD}$

For $C(-2, -4)$ and $D(3, -4)$:
$d_{CD} = \sqrt{(3 - (-2))^2 + (-4 - (-4))^2} = \sqrt{(3 + 2)^2 + 0} = \sqrt{25} = 5$
So, $\overline{AB} \cong \overline{CD}$.

Step 3: Length of $\overline{AC}$

For $A(-2, 2)$ and $C(-2, -4)$:
$d_{AC} = \sqrt{((-2) - (-2))^2 + (-4 - 2)^2} = \sqrt{0 + (-6)^2} = \sqrt{36} = 6$

Step 4: Length of $\overline{BD}$

For $B(3, 2)$ and $D(3, -4)$:
$d_{BD} = \sqrt{((3) - (3))^2 + (-4 - 2)^2} = \sqrt{0 + (-6)^2} = \sqrt{36} = 6$
So, $\overline{AC} \cong \overline{BD}$.

---

3. Conclusion

• Congruent Segments:
  $\overline{AB} \cong \overline{CD}$ (both horizontal sides).
  $\overline{AC} \cong \overline{BD}$ (both vertical sides).

• Explanation:
  These segments are congruent because they have the same length, as calculated using the distance formula.

---

Would you like a detailed graph? Let me know!

---

Questions for Practice:

1. What type of quadrilateral is formed by these points? Why?
2. Are the diagonals of the shape congruent? Prove it using the distance formula.
3. Calculate the slope of each side. What does this tell you about the angles of the quadrilateral?
4. What is the perimeter of the quadrilateral?
5. What is the area of the quadrilateral?

---

Tip: When determining whether sides or diagonals are congruent, always use the distance formula and carefully track each point’s coordinates!