To solve for $x$ in the similar triangles shown in the image, we will use the property of similar triangles: corresponding sides are proportional.

The two triangles $\triangle DBF$ and $\triangle MTX$ are similar since they share the same angles: 37°, 65°, and the remaining angle (since the sum of angles in a triangle is 180°).

Step 1: Set up the proportion of corresponding sides

• $DF$ corresponds to $MX$
• $BF$ corresponds to $TX$
• $DB$ corresponds to $MT$

From the triangles:

• $DF = 20$, $MX = 3$
• $BF = 15$, $TX = x$

Now set up the proportion: $\frac{BF}{TX} = \frac{DF}{MX}$ $\frac{15}{x} = \frac{20}{3}$

Step 2: Solve the proportion

To solve for $x$, cross-multiply: $15 \times 3 = 20 \times x$ $45 = 20x$ Now, divide both sides by 20: $x = \frac{45}{20} = 2.25$

Final answer:

The value of $x$ is $2.25$.

Would you like more details on any step? Here are some questions to explore further:

1. How do we identify similar triangles based on their angles?
2. What are the different methods to prove triangle similarity?
3. Can side lengths of similar triangles be used to find angles?
4. How do the ratios between sides of similar triangles help in real-life applications?
5. What other properties of similar triangles could we use to solve different types of problems?

Tip: Always verify that the triangles are similar by comparing both their angles and their side ratios before setting up proportions.