Given the triangle ABC with side lengths:
- AB = 2x
- BC = 4x - 10
- AC = 3x - 7
and the angles ∠ABC and ∠BCA are congruent, this means that triangle ABC is isosceles with sides AB and AC corresponding to the congruent angles.
Since \( AB \) and \( AC \) are the two sides opposite the equal angles, we can set them equal to each other:
\[ AB = AC \]
Substituting the expressions for the lengths:
\[ 2x = 3x - 7 \]
Next, we'll solve for \( x \):
- First, subtract \( 2x \) from both sides:
\[ 0 = 3x - 2x - 7 \]
This simplifies to:
\[ 0 = x - 7 \]
- Adding 7 to both sides gives us:
\[ x = 7 \]
Now that we have the value of \( x \), we can find the lengths of each side, particularly \( AC \):
\[ AC = 3x - 7 = 3(7) - 7 = 21 - 7 = 14 \]
Thus, the length of line segment \( AC \) is:
\[ \boxed{14} \]
To verify, we can also calculate the lengths of the other sides:
- For \( AB \):
\[ AB = 2x = 2(7) = 14 \]
- For \( BC \):
\[ BC = 4x - 10 = 4(7) - 10 = 28 - 10 = 18 \]
Both \( AB \) and \( AC \) are equal (14), confirming the triangle properties, and \( BC \) is calculated as 18, ensuring all lengths fit our solution.
In conclusion, the length of line segment \( AC \) is:
\[ \boxed{14} \]
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