To solve for $TU$ in this problem, we need to analyze the setup of the triangle and apply proportional reasoning, as segments $IG$ and $UH$ appear to be parallel.

1. Identify given information:

   • $IG$ and $UH$ are marked with matching hash marks, which indicates that they are parallel. This creates two similar triangles: $\triangle IUG$ and $\triangle TUH$.
   • The length $IG$ is given as $x + 2$, and $UH$ is given as $2x - 17$.

2. Set up a proportion: Since the triangles are similar, the ratios of corresponding sides are equal. Therefore: $\frac{IG}{UH} = \frac{IU}{UT}$

3. Substitute known values: From the diagram, we can assign:

   • $IG = x + 2$
   • $UH = 2x - 17$
   • $IU$ and $UT$ are congruent because they have the same markings, meaning $IU = UT$.

   Therefore, we have: $\frac{x + 2}{2x - 17} = 1$

4. Solve for $x$: Multiply both sides by $2x - 17$ to eliminate the denominator: $x + 2 = 2x - 17$ Rearrange to isolate $x$: $2 + 17 = 2x - x$ $x = 19$

5. Find $TU$: Substitute $x = 19$ back into $TU = 2x - 17$: $TU = 2(19) - 17 = 38 - 17 = 21$

So, the length of $TU$ is 21.

Would you like further explanation on any part?

Here are five related questions to expand your understanding:

1. How do we know that triangles $IUG$ and $TUH$ are similar?
2. What would happen if $IU$ and $UT$ were not equal?
3. How can you determine the similarity of triangles in other types of problems?
4. What other methods could solve for unknown sides in similar triangles?
5. How do parallel lines affect proportional relationships in geometry?

Tip: In geometry, similar triangles have corresponding angles equal and their sides in proportion, which can be a powerful tool for finding unknown lengths.