https://file.aiquickdraw.com/mathbot/file/1730131033679v4xh17yr.jpg

To solve for \( x \) in this triangle, let's analyze the given information:

1. **Known Measurements:**
   - The length of the base is \( 30 \).
   - The angle adjacent to this base is split into \( 20^\circ \) and \( 28^\circ \), giving a total angle of \( 48^\circ \) at the base.

2. **Goal:**
   - We want to find the vertical height \( x \) from the top of the triangle to the base.

3. **Approach:**
   This can be solved using trigonometric ratios in right triangles.

Let's break down the problem step-by-step.

### Step 1: Calculate the third angle in the larger triangle.
The larger triangle has:
\[
180^\circ - 90^\circ - 48^\circ = 42^\circ
\]

### Step 2: Apply trigonometric ratios.
In the larger triangle:
- The base (adjacent to \( 48^\circ \)) is \( 30 \).
- The height \( x \) is opposite the \( 42^\circ \) angle.

Using the tangent function:
\[
\tan(42^\circ) = \frac{x}{30}
\]

### Step 3: Solve for \( x \).
Rearrange the equation to find \( x \):
\[
x = 30 \cdot \tan(42^\circ)
\]

Using a calculator:
\[
x \approx 30 \cdot 0.9004 \approx 27.012
\]

### Answer:
The height \( x \approx 27.01 \).

Would you like more details, or have any questions?

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**Related Questions:**
1. What trigonometric function would be used if we needed to find the hypotenuse?
2. How would you solve for \( x \) if the angle were different?
3. How can this setup be used in real-world applications?
4. What are the relationships among all three angles in a right triangle?
5. How would the answer change if the base length were different?

**Tip:** When working with right triangles, remember that the sum of the angles is always \( 180^\circ \).

Learn how to solve for the height x in a right triangle with given base and angles. This problem involves trigonometric concepts, particularly the tangent function, and is suitable for students in Grades 9-10.

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