Solve for X: Similar Triangles with Expressions 6X and 10X-58

To solve the problem, we will perform algebraic manipulation to find $X$.

The triangle gives expressions for sides: $6X$ and $10X - 58$. To find where these are potentially determined equal or prominent in symmetry or division:

• Set the expressions forming these sides equal to each other:

$6X = 10X - 58$

Solve this equation for $X$:

• Subtract $6X$ from both sides:

$0 = 4X - 58$

• Add 58 to both sides:

$58 = 4X$

• Divide both sides by 4 to solve for $X$:

$X = \frac{58}{4}$

Upon simplification:

$X = 14.5$

Therefore, the solution is $X = 14.5$, confirmed as the valid solution satisfying provided problem setup.