Parallel lines BD and EG cut by transversal AH, forming angles of 32°, 148°, and x.

We are given a geometric figure involving two parallel lines and a transversal, with some angles labeled. Let's analyze the image step by step to solve for the unknown angle $ x $.

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Given:

- Lines $ BD $ and $ EG $ are parallel (indicated by the arrows).
- Line $ AH $ is a transversal intersecting both parallel lines.
- At point $ F $, we have:
- One angle marked as $ 32^\circ $
- Another adjacent angle marked as $ 148^\circ $
- At point $ C $, there is an angle marked as $ x $, which is the one we need to find.

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Step 1: Understand the angles at point F

At point $ F $, the transversal $ AH $ crosses line $ EG $. The angles around point $ F $ on the same side of the transversal are:

- $ 32^\circ $
- $ 148^\circ $

These two angles are adjacent and form a straight line (since they lie on line $ EG $), so their sum should be $ 180^\circ $:

$$
32^\circ + 148^\circ = 180^\circ
$$

✔ This checks out — they are supplementary.

Now, notice that the $ 32^\circ $ angle and the $ 148^\circ $ angle are on opposite sides of the transversal. But more importantly, the $ 32^\circ $ angle is between the transversal and the lower line $ EG $, and the $ 148^\circ $ is the other adjacent angle.

But here’s the key: The $ 32^\circ $ angle is an alternate interior angle with respect to the upper line $ BD $.

Wait — let's clarify.

Let’s look at the configuration:

- The transversal $ AH $ intersects:
- Line $ BD $ at point $ C $
- Line $ EG $ at point $ F $
- Since $ BD \parallel EG $, we can use properties of parallel lines and transversals.

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Step 2: Use Alternate Interior Angles

At point $ F $, the angle between the transversal $ AH $ and line $ EG $ is $ 32^\circ $. That is, the interior angle on the left side of the transversal is $ 32^\circ $.

This $ 32^\circ $ angle has a corresponding alternate interior angle on the other side of the transversal, between the upper line $ BD $ and the transversal $ AH $, but on the opposite side.

That alternate interior angle would be located at point $ C $, on the same side of the transversal as the $ 32^\circ $ angle?

Wait — let's think carefully.

Actually, alternate interior angles are on opposite sides of the transversal and inside the parallel lines.

So:

- The $ 32^\circ $ angle at $ F $ is interior, on the left side of the transversal.
- Its alternate interior angle would be on the right side of the transversal, inside the parallel lines, at point $ C $.

But wait — in the diagram, the angle marked $ x $ is at point $ C $, and it's on the top right side of the transversal, outside the parallel lines? Wait — no.

Let’s define the positions:

- Line $ BD $ is above, $ EG $ below.
- Transversal $ AH $ goes from top-left to bottom-right.
- At point $ C $, the angle $ x $ is above line $ BD $, on the right side of the transversal.

But that means $ x $ is not an interior angle — it's an exterior angle.

Wait — perhaps we need to consider corresponding angles or supplementary angles.

Let’s try another approach.

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Step 3: Use Vertical Angles and Supplementary Angles

At point $ F $, we know:

- The angle between the transversal and line $ EG $ is $ 32^\circ $.
- The adjacent angle is $ 148^\circ $, which is supplementary: $ 32^\circ + 148^\circ = 180^\circ $.

Now, the $ 32^\circ $ angle is interior, and since $ BD \parallel EG $, the corresponding angle at point $ C $ should be equal.

But where is the corresponding angle?

- The $ 32^\circ $ angle is on the left side of the transversal, below line $ EG $.
- Its corresponding angle would be on the left side of the transversal, above line $ BD $.

But the angle marked $ x $ is on the right side of the transversal.

So $ x $ is not corresponding to $ 32^\circ $.

Wait — perhaps $ x $ is supplementary to the corresponding angle.

Let’s look at the angles at point $ C $.

At point $ C $, the transversal $ AH $ crosses line $ BD $. The angle $ x $ is the angle above the line $ BD $, on the right side of the transversal.

That means $ x $ is an exterior angle.

But what is the interior angle at $ C $ on the left side?

Let’s suppose that the interior angle at $ C $, on the left side of the transversal, is equal to the $ 32^\circ $ angle at $ F $, because they are alternate interior angles.

So:

> Alternate interior angles: $ \angle $ at $ F $ (32°) = $ \angle $ at $ C $ (on the left side, interior)

So, the interior angle at $ C $ on the left is $ 32^\circ $.

Then, the angle $ x $ is adjacent to this $ 32^\circ $ angle, forming a straight line along $ BD $.

So:

$$
x + 32^\circ = 180^\circ
$$

$$
x = 180^\circ - 32^\circ = 148^\circ
$$

Wait — but that seems too large? Let's check.

No — actually, if the interior angle at $ C $ is $ 32^\circ $, then the exterior angle adjacent to it (on the same side of the transversal) would be $ 180^\circ - 32^\circ = 148^\circ $.

But in the diagram, the angle $ x $ is shown as being on the opposite side of the transversal — wait.

Let’s re-express:

At point $ C $, the transversal splits the line $ BD $. The angles around point $ C $ are:

- One angle is $ x $, shaded pink, above $ BD $, to the right of $ AH $
- The angle below $ BD $, to the left of $ AH $, is $ 32^\circ $'s alternate interior angle → $ 32^\circ $

Wait — no. Actually, the $ 32^\circ $ angle is at $ F $, below $ EG $, on the left of the transversal.

Its alternate interior angle is at $ C $, above $ BD $, on the right of the transversal? No — alternate interior angles are on opposite sides of the transversal and between the lines.

So:

- At $ F $: $ 32^\circ $ is between the lines, on the left of the transversal.
- Then, the alternate interior angle at $ C $ is between the lines, on the right of the transversal.

So at point $ C $, the interior angle on the right side of the transversal is $ 32^\circ $.

But in the diagram, the angle $ x $ is above $ BD $, on the right side of the transversal — so it's not interior; it's exterior.

Wait — but if $ BD $ is the upper line, and $ AH $ cuts through it at $ C $, then the region above $ BD $ is outside the parallel lines.

So the angle $ x $ is not between the lines — it's outside.

So $ x $ is an exterior angle.

But the angle between the lines at $ C $, on the right side of the transversal, is $ 32^\circ $ (by alternate interior angles).

Then, the angle $ x $ is adjacent to that $ 32^\circ $ angle, forming a straight line along $ BD $.

So:

$$
x + 32^\circ = 180^\circ \Rightarrow x = 148^\circ
$$

Yes! So $ x = 148^\circ $

But wait — in the diagram, at point $ F $, there's already a $ 148^\circ $ angle. Is that a coincidence?

Let’s verify.

At $ F $, the angle $ 148^\circ $ is adjacent to the $ 32^\circ $ angle, and together they make $ 180^\circ $ — correct.

But the $ 148^\circ $ angle at $ F $ is exterior to the parallel lines — it's on the right side of the transversal, below $ EG $.

Its corresponding angle would be at $ C $, on the right side of the transversal, above $ BD $ — which is exactly where $ x $ is!

Ah! So $ x $ and the $ 148^\circ $ angle at $ F $ are corresponding angles.

Since $ BD \parallel EG $, corresponding angles are equal.

Therefore:

$$
x = 148^\circ
$$

✔ This matches.

Alternatively, using alternate interior angles:

- $ 32^\circ $ at $ F $ corresponds to the interior angle at $ C $ on the right side → $ 32^\circ $
- Then $ x $ is supplementary to that: $ x = 180^\circ - 32^\circ = 148^\circ $

Same result.

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✔ Final Answer:

$$
\boxed{x = 148^\circ}
$$

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Explanation Summary:

- Lines $ BD $ and $ EG $ are parallel.
- Transversal $ AH $ intersects them.
- The $ 32^\circ $ angle at $ F $ has a corresponding interior angle at $ C $ (on the right) of $ 32^\circ $.
- The angle $ x $ is adjacent to this $ 32^\circ $ angle on line $ BD $, so they are supplementary.
- Therefore, $ x = 180^\circ - 32^\circ = 148^\circ $.
- Alternatively, $ x $ and the $ 148^\circ $ angle at $ F $ are corresponding angles, so they are equal.

Thus, $ \boxed{x = 148^\circ} $.