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To find the value of \( x \) in the given triangles, we will use the properties of similar triangles.

The two triangles \( \triangle DBF \) and \( \triangle TMX \) are given to have the following angles:
- \( \angle D = \angle T = 37^\circ \)
- \( \angle F = \angle X = 65^\circ \)
- Therefore, \( \angle B = \angle M = 180^\circ - 37^\circ - 65^\circ = 78^\circ \)

Since all the corresponding angles are equal, the triangles are similar by the Angle-Angle (AA) similarity criterion. For similar triangles, the ratios of corresponding sides are equal. Therefore, we can write:

\[
\frac{DF}{TX} = \frac{BD}{TM} = \frac{BF}{MX}
\]

From the triangles, we have:
- \( DF = 15 \)
- \( TX = 3 \)
- \( BD = 20 \)
- \( TM = x \)

Using the ratio of the sides corresponding to \( BD \) and \( TM \):

\[
\frac{BD}{TM} = \frac{20}{x}
\]

Similarly, using the ratio of the sides corresponding to \( DF \) and \( TX \):

\[
\frac{DF}{TX} = \frac{15}{3} = 5
\]

Therefore, since the triangles are similar, we can write:

\[
\frac{20}{x} = 5
\]

Solving for \( x \):

\[
20 = 5x
\]
\[
x = \frac{20}{5} = 4
\]

So, the value of \( x \) is \( 4 \).

Would you like more details or have any questions about this solution?

Here are five related questions to expand your understanding:
1. How do you prove that two triangles are similar?
2. What are the criteria for two triangles to be similar?
3. How can you find the lengths of other sides in similar triangles?
4. What is the significance of the Angle-Angle (AA) similarity criterion?
5. How does the ratio of corresponding sides in similar triangles help in solving real-world problems?

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

Learn how to find the value of x in similar triangles using the Angle-Angle similarity criterion. This solution explains how the ratio of corresponding sides helps determine the unknown side, suitable for students in Grades 8-10.

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