1. **State the problem:** We are given a triangle with a mid-segment parallel to the base. The top side is labeled $4x$, the mid-segment is labeled 14, one side is labeled $7x - 9$, and the base is labeled 14. We need to find the value of $x$ using the converse of the mid-segment theorem. 2. **Recall the mid-segment theorem:** The mid-segment theorem states that the segment connecting the midpoints of two sides of a triangle is parallel to the third side and its length is half the length of that third side. 3. **Converse of the mid-segment theorem:** If a segment is parallel to one side of a triangle and its length is half the length of that side, then it connects the midpoints of the other two sides. 4. **Apply the theorem to the problem:** The mid-segment is given as 14, and the base is also 14. Since the mid-segment should be half the length of the base, but here both are equal, this suggests the mid-segment is not half but equal to the base. This implies the segment labeled 14 is not a mid-segment unless the triangle is degenerate or the labeling corresponds differently. 5. **Check the given segments:** The top side is $4x$, the mid-segment is 14, and one side is $7x - 9$. Since the mid-segment is parallel to the base (14), by the converse theorem, the mid-segment length should be half the base length. 6. **Set up the equation:** Since the mid-segment is 14 and the base is 14, the mid-segment should be half the base, so: $$14 = \frac{1}{2} \times 14$$ This is false, so instead, the problem likely wants us to use the equality of the two segments labeled 14 to find $x$. 7. **Use the equality of the two segments labeled 14:** The mid-segment is 14, and the base is 14, so the mid-segment equals the base. This suggests the segment labeled $4x$ is equal to the mid-segment 14, and the segment labeled $7x - 9$ is equal to the base 14. 8. **Set up equations:** $$4x = 14$$ $$7x - 9 = 14$$ 9. **Solve the first equation:** $$4x = 14$$ $$x = \frac{14}{4} = 3.5$$ 10. **Check the second equation with $x=3.5$:** $$7(3.5) - 9 = 24.5 - 9 = 15.5 \neq 14$$ 11. **Since the second equation does not equal 14, check if the mid-segment theorem applies to the sides $4x$ and $7x - 9$:** The mid-segment theorem states the mid-segment is half the base, so: $$14 = \frac{1}{2} \times (7x - 9)$$ 12. **Solve for $x$:** $$14 = \frac{7x - 9}{2}$$ Multiply both sides by 2: $$28 = 7x - 9$$ Add 9 to both sides: $$37 = 7x$$ Divide both sides by 7: $$x = \frac{37}{7} \approx 5.2857$$ 13. **Verify $4x$ with this $x$:** $$4 \times \frac{37}{7} = \frac{148}{7} \approx 21.14$$ This does not equal 14, so the segment labeled $4x$ is not equal to the mid-segment. 14. **Conclusion:** The value of $x$ that satisfies the mid-segment theorem using the side $7x - 9$ and the mid-segment 14 is: $$\boxed{\frac{37}{7}}$$