A quadrilateral is cut from a piece of gift wrapping paper, which has equally wide white and gray stripes.
The grey stripes in the quadrilateral have a combined area of $10$. Determine the area of the quadrilateral.
[img]https://1.bp.blogspot.com/-ia13b4RsNs0/XzP0cepAcEI/AAAAAAAAMT8/0UuCogTRyj4yMJPhfSK3OQihRqfUT7uSgCLcBGAsYHQ/s0/2020%2Bmohr%2Bp2.png[/img]
1. **Assume the quadrilateral is a rectangle:**
   - Let's assume the quadrilateral is a rectangle with sides parallel or orthogonal to the stripes. This assumption simplifies the problem because the area of the stripes within the quadrilateral will be proportional to the total area of the quadrilateral.

2. **Calculate the area of the grey stripes:**
   - Given that the grey stripes have a combined area of $10$, and knowing that the stripes are equally wide, we can infer that the white stripes also have an equal area within the quadrilateral. Therefore, the total area of the quadrilateral is twice the area of the grey stripes.

3. **Determine the total area of the quadrilateral:**
   - Since the grey stripes cover an area of $10$, and the white stripes cover an equal area, the total area of the quadrilateral is:
   \[
   \text{Total area} = \text{Area of grey stripes} + \text{Area of white stripes} = 10 + 10 = 20
   \]

4. **Verify the rectangular assumption:**
   - To ensure that the rectangular assumption is sufficient, consider shifting the vertices of the quadrilateral along the edges of the stripes. This shift does not change the area of the quadrilateral. By transforming the quadrilateral into a triangle and then doubling it to form a rectangle, we confirm that the area calculation remains valid.

The final answer is $\boxed{20}$