In the given image, we are asked to find the values of $t$ and $u$ in the triangle geometry.

Let's break it down step by step:

1. Identifying key elements:

   • Triangle $HIG$ is isosceles because two sides are marked as equal (between $H$ and $I$, and between $G$ and $F$).
   • Angle $\angle HIG = t$.
   • We are given $\angle H = 36^\circ$ and $HG = 2$.

2. Using properties of isosceles triangles:

   • Since triangle $HIG$ is isosceles, the base angles, which are angles $\angle GIH$ and $\angle GHI$, are equal.
   • Let these equal angles be $\theta$.
   • Using the angle sum property of a triangle ($\angle H + 2\theta = 180^\circ$) in $\triangle HIG$, we can find $\theta$ and ultimately $t$.

3. Finding angle $t$:

   • Since $\triangle HIG$ is isosceles, we can apply the formula: $\text{Sum of angles in a triangle} = 180^\circ$ $t + 36^\circ + 36^\circ = 180^\circ$ Solving for $t$: $t = 180^\circ - 72^\circ = 108^\circ$

4. Finding $u$:

   • Next, we need to find angle $u$. Since $\triangle GIF$ is also isosceles (as indicated by the two marked sides), we apply the same method for finding the base angles.
   • Assuming $\triangle GIF$ is isosceles, the remaining angles can be calculated based on angle properties and side lengths, but without side lengths, we assume base angles are equal, simplifying the case.

Thus:

• $t = 108^\circ$
• $u$ will depend on further provided geometric context (side lengths, specific triangle properties).

Would you like further clarification or additional details on any steps?

Here are 5 related questions:

1. What properties of isosceles triangles are useful in solving for unknown angles?
2. How does the angle sum property help in solving triangle problems?
3. Can we apply trigonometric ratios to find unknown lengths in this triangle?
4. What happens if one angle of a triangle is greater than $90^\circ$? What type of triangle is it?
5. Why are both the base angles of an isosceles triangle equal?

Tip: Always remember the sum of the angles in any triangle is always $180^\circ$, a fundamental principle in triangle geometry.