Hexagon with algebraic side lengths and two 50-degree angles.

It looks like you've uploaded an image of a geometric figure — specifically, a hexagon with some angles and side expressions labeled. Let's analyze the problem step by step.

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🔍 Given:

We have a hexagon with:

- Two angles marked as $50^\circ$.
- Four sides labeled with algebraic expressions:
- Top side: $11x - 30$
- Right side: $5x$
- Bottom side: $2x + 60$
- Left side: $6x - 10$

Also, there are red lines extending from the vertices, possibly indicating that the hexagon is inscribed between parallel lines or has parallel sides, but more importantly, the angles at two vertices are given as $50^\circ$.

But wait — in a regular hexagon, all internal angles are $120^\circ$, so this is not regular. But we're given only two angles as $50^\circ$, which seems small for an internal angle.

Wait — let’s reconsider: Are those $50^\circ$ internal angles? Or external angles?

Looking at the diagram (as described), the $50^\circ$ angles are shown between the red lines and the sides — and the red lines seem to be extensions of the sides.

This suggests that the $50^\circ$ angles are external angles, or perhaps adjacent to the interior angles.

But here's a key insight:

> In many such problems, especially when multiple sides are expressed in terms of $x$, and angles are given, the figure may be symmetrical or have opposite sides equal.

Let’s assume this is a convex hexagon with opposite sides equal or parallel, and maybe it's symmetric.

Alternatively, consider that the sum of the interior angles of a hexagon is:

$$
\text{Sum} = (n - 2) \times 180^\circ = (6 - 2) \times 180 = 720^\circ
$$

But we only know two angles: $50^\circ$ each. That’s not enough unless we can infer more.

Wait — but look again: the $50^\circ$ angles are not inside the polygon; they are outside, formed between the extensions of the sides and the sides themselves.

So these might be external angles.

But if the red lines are extensions of the sides, then the $50^\circ$ angles are likely supplementary to the internal angles.

Let’s suppose that the red lines are extensions of the adjacent sides, forming exterior angles of $50^\circ$.

But in a polygon, exterior angles add up to $360^\circ$. If two exterior angles are $50^\circ$, that’s only $100^\circ$, so others must make up the rest.

But without knowing how many such angles exist, this might not help directly.

Alternatively, notice that the side lengths are labeled in terms of $x$, and we might need to solve for $x$ using symmetry or equality of opposite sides.

Let’s try that idea.

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✔ Key Observation:

The figure appears to be a symmetrical hexagon, possibly equiangular or equilateral in parts, and the side labels suggest that opposite sides may be equal.

Let’s label the sides:

- Top: $11x - 30$
- Right: $5x$
- Bottom: $2x + 60$
- Left: $6x - 10$

Wait — there are six sides, but only four are labeled.

Let me count:

From the description and typical diagrams:

- Top side: $11x - 30$
- Right side: $5x$
- Bottom side: $2x + 60$
- Left side: $6x - 10$

But there are six sides, so two are missing.

Wait — actually, looking at the shape:

It appears to be a hexagon with alternating long and short sides, and the red lines suggest symmetry.

Possibly, the hexagon has three pairs of equal opposite sides?

Or perhaps it's a parallelogram-like shape with symmetry.

Another possibility: the red lines indicate that some sides are parallel, and the angles between them are $50^\circ$.

But here’s a better idea:

Let’s suppose that the hexagon is symmetrical, and opposite sides are equal.

Then:

- Top side: $11x - 30$
- Bottom side: $2x + 60$

If they are opposite, maybe they are equal?

Similarly, left and right sides: $6x - 10$ and $5x$

But that would give:

$$
11x - 30 = 2x + 60 \quad \text{(top = bottom)}
$$

Solve:

$$
11x - 30 = 2x + 60 \\
11x - 2x = 60 + 30 \\
9x = 90 \\
x = 10
$$

Now check other sides:

Left: $6x - 10 = 6(10) - 10 = 60 - 10 = 50$

Right: $5x = 5(10) = 50$

So both left and right are $50$ — equal!

So opposite sides are equal.

Now check top: $11x - 30 = 110 - 30 = 80$

Bottom: $2x + 60 = 20 + 60 = 80$ — also equal.

So all opposite sides are equal.

That makes sense.

Now, what about the angles?

We were given two $50^\circ$ angles — likely exterior angles or angles between extensions and sides.

But now that we’ve found $x = 10$, we can verify consistency.

But is there another way to confirm?

Let’s see: if the hexagon has equal opposite sides, and the angles are symmetric, then $x = 10$ is consistent.

But could there be another condition?

Wait — maybe the $50^\circ$ angles are interior angles, but that seems unlikely because $50^\circ$ is too small for a convex hexagon — typically interior angles are greater than $90^\circ$.

But perhaps it's a non-convex hexagon?

Alternatively, the $50^\circ$ angles might be between the sides and the red lines, which are extensions, so they could be exterior angles.

But if two exterior angles are $50^\circ$, and the hexagon is symmetric, perhaps others are equal.

But we don’t have enough info.

However, since setting opposite sides equal gives us a consistent value of $x = 10$, and all side lengths become:

- Top: $11(10) - 30 = 80$
- Right: $5(10) = 50$
- Bottom: $2(10) + 60 = 80$
- Left: $6(10) - 10 = 50$

So opposite sides are equal: top-bottom = 80, left-right = 50.

This strongly suggests symmetry.

Moreover, the two $50^\circ$ angles are on opposite sides, which fits symmetry.

Therefore, the most logical conclusion is that the figure is symmetric, and opposite sides are equal, leading to:

$$
11x - 30 = 2x + 60
$$

Solving:

$$
11x - 2x = 60 + 30 \\
9x = 90 \\
x = 10
$$

✔ Answer: $x = 10$

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📌 Final Answer:

$$
\boxed{x = 10}
$$

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✔ Explanation:

- The hexagon appears to have symmetry, with opposite sides equal.
- Setting the top side ($11x - 30$) equal to the bottom side ($2x + 60$) gives the equation:
$$
11x - 30 = 2x + 60
$$
- Solving yields $x = 10$.
- Verification shows that the left and right sides are also equal when $x = 10$: $6x - 10 = 50$, $5x = 50$.
- Thus, the solution is consistent with the symmetry of the figure.

So, the value of $x$ is $\boxed{10}$.