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We are given two triangles and need to solve for \( x \), the side in the second triangle. The first triangle on the left provides necessary side and angle information.

### Step 1: Identify Similarity
Notice that both triangles have two angles of 55° each. This suggests that the triangles are similar by the **Angle-Angle (AA)** similarity criterion.

### Step 2: Set Up Proportions
Since the triangles are similar, corresponding sides are proportional. The known side lengths of the first triangle are \( 19 \) and \( 20 \), and the known side in the second triangle is \( 30 \), while \( x \) is the unknown side corresponding to \( 19 \).

Thus, we can set up the following proportion:
\[
\frac{19}{x} = \frac{20}{30}
\]

### Step 3: Solve for \( x \)
We can cross-multiply to solve for \( x \):
\[
19 \times 30 = 20 \times x
\]
\[
570 = 20x
\]
\[
x = \frac{570}{20} = 28.5
\]

### Final Answer:
The value of \( x \) is \( 28.5 \).

Would you like further details or have any additional questions?

Here are some related questions to deepen your understanding:
1. How do we identify if two triangles are similar?
2. Why is the AA criterion sufficient for triangle similarity?
3. What other methods can be used to solve for unknown sides in similar triangles?
4. Can similarity be applied to non-triangular figures?
5. How do different triangle theorems (e.g., Pythagoras) relate to similarity?

**Tip:** When working with similar triangles, always compare corresponding angles and sides carefully to set up correct proportions.

Solve for x in the given two triangles with corresponding angles and sides. One triangle has angles 55°, 61°, and the side lengths 19 and 20. The second triangle has angles 55°, 64°, and the known side length 30, with the side x to be solved for.

Learn how to solve for the unknown side x in two similar triangles using the AA similarity criterion. This problem involves setting up proportions between corresponding sides of the triangles. Suitable for students in Grades 8-10.

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