To solve the problem in the image, let's analyze the geometry and use the midsegment theorem:

Problem Analysis

The triangle $JKL$ is formed by connecting the midpoints of the sides of triangle $GHI$.
The midsegment theorem states that:

1. A midsegment in a triangle is parallel to the third side of the triangle.
2. The midsegment divides the triangle into two smaller triangles that are similar to the original triangle.

Steps to Solve

1. Identify $\angle GLJ$: We are tasked with finding the measure of $\angle GLJ$.
   By examining the angles provided:

   • In $\triangle GHI$:
     • $\angle G = 46^\circ$
     • $\angle H = 97^\circ$
     • $\angle I = 37^\circ$
   • The midsegment divides $\triangle GHI$ such that corresponding angles in $\triangle JKL$ are the same as in $\triangle GHI$ due to similarity.

2. Use the Triangle Sum Property:

   • The sum of angles in a triangle is $180^\circ$. For $\triangle GHI$, we verify: $\angle G + \angle H + \angle I = 46^\circ + 97^\circ + 37^\circ = 180^\circ$

3. Corresponding Angles in $\triangle JKL$:

   • Since $JKL$ is similar to $GHI$, the angles in $JKL$ correspond directly to the angles in $GHI$:
     • $\angle GLJ = \angle G = 46^\circ$.

Final Answer:

$m\angle GLJ = 46^\circ$

Would you like further clarification or assistance with similar problems?

Related Questions:

1. Why does the midsegment theorem ensure similarity between triangles?
2. How can you prove that a midsegment is parallel to the third side of the triangle?
3. What are some other properties of midsegments in triangles?
4. Can you find the lengths of the sides of $\triangle JKL$ given the dimensions of $\triangle GHI$?
5. How does the triangle sum property relate to midsegment problems?

Tip:

When working with similar triangles, always look for corresponding angles and use proportionality for sides if needed!