We are given a circle with chords $EI$ and $FG$ intersecting inside the circle. We are given the measure of arc $EI$ is 65 degrees, and we need to find the measure of angle $x$, which is angle $EFI$. Since $GHI$ appears to be a diameter (although it is not explicitly stated), the center of the circle is on this segment.

1. Problem Description

We are given a circle with chords $EI$ and $FG$ intersecting inside the circle. We are given the measure of arc $EI$ is 65 degrees, and we need to find the measure of angle $x$, which is angle $EFI$. Since $GHI$ appears to be a diameter (although it is not explicitly stated), the center of the circle is on this segment.

2. Solution Steps

We are looking for $x = \angle EFI$. Since $\angle EFI$ is an inscribed angle, its measure is half the measure of the intercepted arc $FI$. So, $x = \frac{1}{2} m(FI)$.

We know that $m(EI) = 65^\circ$. If $GHI$ is a diameter, then $GH$ is a radius, and therefore $H$ is the center of the circle. If $GHI$ is a diameter, then the arc $GI$ has the same measure as the angle $GHI$ subtends from the circle. However, the picture is not precise enough to confirm whether $GHI$ is a diameter.

However, $\angle EFI$ is an inscribed angle that intercepts arc $FI$. $\angle EGI$ is also an inscribed angle that intercepts arc $EI$. The measure of an inscribed angle is half the measure of its intercepted arc. Therefore,

$$m\angle EGI = \frac{1}{2} m(\text{arc } EI) = \frac{1}{2}(65^\circ) = 32.5^\circ$$

Since the inscribed angles $\angle EFI$ and $\angle EGI$ intercept the same arc $EI$, $\angle EFI$ is half of arc $EI$. This is incorrect, the angles $\angle EFI$ and $\angle EGI$ don't intercept the same arc.

It looks as if $\angle EFI$ intercepts the arc $FI$.

If the question is asking for the measure of angle $EFI$ which is labeled as $x$, and if it is a central angle that intercepts the arc $FI$, then it should have the same measure of that arc. However, it is not a central angle. Instead, the angle $EFI$ is an inscribed angle which intercepts the arc $FI$.

Also, we need to assume that arc $GI$ is equal to arc $FI$ to obtain a solution. But there is no information indicating $GI=FI$.

$\angle FEI$ intercepts arc $FI$, and $\angle EFI$ intercepts arc $EI$.

It appears that the question wants you to assume that arc $FI$ has the same measure as angle $EFI$, which would mean that $FI=x$.

In this case, $x = \frac{1}{2}(m(\text{arc }FI))$, where $m(\text{arc }FI)$ is the measure of arc $FI$.

The measure of the inscribed angle EFI is half the measure of the intercepted arc FI.

$$x = \frac{1}{2} m(FI)$$

Also, assuming that the arc GI and the arc FI have the same measure since the lines EG and FI are parallel, then $m(FI) = m(GI)$.

However, the problem does not provide enough information to determine the value of $x$.

If we assumed arc $FI$ had the same measure as arc $EI$, then:

$$x = \frac{1}{2}m(FI) = \frac{1}{2}(65) = 32.5$$

However, we can't assume that they are the same.

3. Final Answer

There is not enough information provided to solve for x.