We are given that $\overline{HI} \cong \overline{TU}$ and $\overline{HJ} \cong \overline{TV}$. We need to prove that $\overline{IJ} \cong \overline{UV}$.

1. Problem Description

We are given that $\overline{HI} \cong \overline{TU}$ and $\overline{HJ} \cong \overline{TV}$. We need to prove that $\overline{IJ} \cong \overline{UV}$.

2. Solution Steps

We know that the length of $\overline{HJ}$ is the sum of the lengths of $\overline{HI}$ and $\overline{IJ}$. Also, the length of $\overline{TV}$ is the sum of the lengths of $\overline{TU}$ and $\overline{UV}$. We can write these relationships as equations:

$HJ = HI + IJ$

$TV = TU + UV$

We are given that $HI = TU$ and $HJ = TV$. Therefore, we can substitute these values into the second equation:

$HJ = HI + UV$

Since $HJ = HI + IJ$, we can substitute $HI + IJ$ for $HJ$:

$HI + IJ = HI + UV$

Now, we can subtract $HI$ from both sides of the equation:

$HI + IJ - HI = HI + UV - HI$

$IJ = UV$

Since $IJ = UV$, we can say that $\overline{IJ} \cong \overline{UV}$.

3. Final Answer

$\overline{IJ} \cong \overline{UV}$