The image shows a triangle configuration where the values of $s$ and $t$ are to be determined. Here's a breakdown of the information provided:

1. Triangle Structure:

   • $\triangle HGI$ is the upper triangle.
   • $\triangle FGI$ is the lower triangle.
   • The angle at $H$ is $66^\circ$.
   • Segment $GI$ and $FG$ are marked as equal (shown with tick marks), implying that $\triangle FGI$ is isosceles.
   • Side $FG = 8$.

2. Isosceles Triangle: Since $FG = GI$, this means $\triangle FGI$ is isosceles, so the angles at $F$ and $I$ are equal, making angle $FGI$ (or $s$) congruent to angle $GFI$.

Plan to Solve:

Step 1: Use angle properties of the isosceles triangle.

• In $\triangle FGI$, the sum of angles must equal $180^\circ$.
• Let the two equal angles be $s$. So, we have: $s + s + \text{angle at } G = 180^\circ$
• To find angle $G$, we can calculate the remaining interior angle of triangle $HGI$ (since we already know angle $H = 66^\circ$).

Step 2: Use angle properties in $\triangle HGI$.

• Sum of interior angles in $\triangle HGI$ must also equal $180^\circ$.
• The known angle at $H = 66^\circ$, so we can calculate the angle at $G$ using: $\text{Angle at } G + \text{Angle at } I = 180^\circ - 66^\circ$
• The angle at $I$ is $t$, and that will help us find $t$.

Let me calculate these values to determine $s$ and $t$.The values of the angles $s$ and $t$ are as follows:

• $s = 61.5^\circ$
• $t = 57.0^\circ$

These values come from using the properties of isosceles triangles and the sum of interior angles in triangles.

Would you like more details on how these were derived, or do you have any further questions?

Here are 5 related questions:

1. How can the exterior angle theorem be applied in this problem?
2. What is the role of the isosceles triangle in solving for $s$?
3. How would the solution change if $FG \neq GI$?
4. Can you calculate the area of $\triangle FGI$ given additional information about the height?
5. How do you verify the sum of interior angles in a triangle?

Tip: Always remember that the sum of interior angles of any triangle is $180^\circ$.