To solve for $x$ in this problem, we can apply the Triangle Proportionality Theorem. According to this theorem, when a line is parallel to one side of a triangle and intersects the other two sides, it divides those sides proportionally.

Given Information:

• The segments on the triangle's sides are divided as follows:
  • The entire bottom segment is $143$.
  • The parallel line divides the bottom segment into two parts, $10x + 6$ and $143$.
  • The parallel line divides the corresponding side lengths into $78$ and $42$.

Steps to Solve:

1. By the Triangle Proportionality Theorem, we can set up a proportion: $\frac{10x + 6}{143} = \frac{78}{42}$

2. Simplify $\frac{78}{42}$:

   • Divide both the numerator and denominator by their greatest common divisor (6): $\frac{78}{42} = \frac{13}{7}$

3. Substitute back into the equation: $\frac{10x + 6}{143} = \frac{13}{7}$

4. Cross-multiply to solve for $x$: $7(10x + 6) = 13 \cdot 143$

5. Expand and calculate: $70x + 42 = 1859$

6. Subtract 42 from both sides: $70x = 1817$

7. Divide by 70 to isolate $x$: $x = \frac{1817}{70} = 26$

Thus, the correct answer is $\text{A) 13}$.