We are given two right triangles that share a common angle and need to find the value of $y$. The triangles are similar, as they have corresponding angles that are equal.

Step 1: Set up a proportion based on corresponding sides.

Since the triangles are similar, the corresponding sides are proportional. The sides corresponding to the angle marked in both triangles are:

$\frac{32}{y + 3} = \frac{20}{25}$

Step 2: Solve the proportion.

To solve for $y$, we cross-multiply:

$32 \times 25 = 20 \times (y + 3)$

$800 = 20(y + 3)$

Step 3: Simplify the equation.

Expand the right side:

$800 = 20y + 60$

Now, subtract 60 from both sides:

$740 = 20y$

Step 4: Solve for $y$.

Divide both sides by 20:

$y = \frac{740}{20} = 37$

Thus, the value of $y$ is $37$.

Would you like more details or have any questions?

Related Questions:

1. How do you determine if two triangles are similar?
2. What is the difference between corresponding angles and corresponding sides?
3. How can the Pythagorean theorem be applied to find missing sides in right triangles?
4. What are the other ways to solve similar triangle problems?
5. How do you check if your proportionality setup is correct?

Tip:

When working with similar triangles, always ensure you correctly identify corresponding sides before setting up a proportion.