The problem asks to show that the size of the angle $FGH$ is $65^\circ$, correct to the nearest degree. We are provided with previous calculations where $IGF1 = \sqrt{10.96} = 3.31$ (approximately).

1. Problem Description

The problem asks to show that the size of the angle $FGH$ is $65^\circ$, correct to the nearest degree. We are provided with previous calculations where $IGF1 = \sqrt{10.96} = 3.31$ (approximately).

2. Solution Steps

Without additional context or a diagram, it is impossible to definitively show that the angle $FGH$ is $65^\circ$. I need more information on the geometric relationship between the points $F, G, H,$ and any other points or lengths given in the problem. The previous calculation $IGF1 = \sqrt{10.96} \approx 3.31$ suggests that this value could be a side length, area, or other parameter that can be used to determine $\angle FGH$.

However, let's consider an example scenario using the Cosine Rule if we had the side lengths of the triangle $FGH$. Let $FG=a$, $GH=b$, and $FH=c$. Then the Cosine Rule states:

$c^2 = a^2 + b^2 - 2ab \cos(G)$

Therefore,

$\cos(G) = \frac{a^2 + b^2 - c^2}{2ab}$

If we are given the lengths of the three sides of the triangle, then we can substitute in the known values to calculate $\cos(G)$ and then use the inverse cosine function ($\cos^{-1}$) to find the angle $G = \angle FGH$.

Without a specific diagram and the lengths of relevant sides, I cannot proceed.

Let me assume that we are given $FG=3$, $GH=4$, and $FH=5.3$. In this case, we would have:

$\cos(\angle FGH) = \frac{3^2 + 4^2 - 5.3^2}{2(3)(4)} = \frac{9 + 16 - 28.09}{24} = \frac{-3.09}{24} = -0.12875$

$\angle FGH = \cos^{-1}(-0.12875) \approx 97.4^\circ$. This is incorrect.

Let's assume we are given $FG=4.5$, $GH=3.2$, and $FH=4$.

$\cos(\angle FGH) = \frac{4.5^2 + 3.2^2 - 4^2}{2(4.5)(3.2)} = \frac{20.25 + 10.24 - 16}{28.8} = \frac{14.49}{28.8} \approx 0.503125$

$\angle FGH = \cos^{-1}(0.503125) \approx 59.8^\circ$ (close to $60^\circ$).

Again, I cannot show that the angle is exactly $65^\circ$ without more information.

3. Final Answer

Insufficient information to solve the problem. More context or a diagram is needed.