A trapezoid has bases with lengths equal to $5$ and $15$ and legs with lengths equal to $13$ and $13.$ Determine the area of the trapezoid.
1. **Identify the given values:**
   - The lengths of the bases of the trapezoid are \( a = 5 \) and \( b = 15 \).
   - The lengths of the legs of the trapezoid are both \( c = 13 \).

2. **Determine the height of the trapezoid:**
   - Since the legs are equal, the trapezoid is isosceles.
   - Drop perpendiculars from the endpoints of the shorter base to the longer base, forming two right triangles and a rectangle in the middle.
   - The length of the rectangle's base is \( 15 - 5 = 10 \).
   - Each right triangle has a base of \( \frac{10}{2} = 5 \) and a hypotenuse of \( 13 \).

3. **Use the Pythagorean Theorem to find the height \( h \):**
   \[
   h = \sqrt{c^2 - \left(\frac{b-a}{2}\right)^2} = \sqrt{13^2 - 5^2}
   \]
   \[
   h = \sqrt{169 - 25} = \sqrt{144} = 12
   \]

4. **Calculate the area of the trapezoid using the area formula:**
   \[
   A = \frac{1}{2} \cdot (a + b) \cdot h
   \]
   \[
   A = \frac{1}{2} \cdot (5 + 15) \cdot 12
   \]
   \[
   A = \frac{1}{2} \cdot 20 \cdot 12
   \]
   \[
   A = 10 \cdot 12 = 120
   \]

The final answer is \(\boxed{120}\).