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The diagram shows a pair of triangles, and they seem to be in proportion as they are connected by a transversal at point Z. We will solve for \( X \) and \( Y \) using the concept of similar triangles, which says that the corresponding sides of similar triangles are proportional.

Given:
- \( WZ = 2y + 15 \)
- \( ZY = 9y - 13 \)
- \( WX = x + 5 \)
- \( XY = 3x - 5 \)

Since these triangles are similar, we can set up the following proportion:

\[
\frac{WZ}{ZY} = \frac{WX}{XY}
\]

Substitute the given expressions into this proportion:

\[
\frac{2y + 15}{9y - 13} = \frac{x + 5}{3x - 5}
\]

Now, we have a system of equations that we can solve to find the values of \( x \) and \( y \).

### Step 1: Cross Multiply

\[
(2y + 15)(3x - 5) = (x + 5)(9y - 13)
\]

We will now expand both sides.

### Left Side:

\[
(2y + 15)(3x - 5) = 6xy - 10y + 45x - 75
\]

### Right Side:

\[
(x + 5)(9y - 13) = 9xy - 13x + 45y - 65
\]

### Step 2: Equate the two sides

\[
6xy - 10y + 45x - 75 = 9xy - 13x + 45y - 65
\]

Now, simplify this equation.

### Step 3: Combine Like Terms

Move all the terms to one side to simplify:

\[
6xy - 9xy + 45x + 13x - 10y - 45y - 75 + 65 = 0
\]

This simplifies to:

\[
-3xy + 58x - 55y - 10 = 0
\]

This is one equation, but since we have two variables (\(x\) and \(y\)), we may need more information or another condition to solve for both variables.

Would you like to continue with solving this equation, or do you have further details from the problem (perhaps related to "Blank 1" and "Blank 2")? 

Feel free to ask for any clarification.

---

Here are 5 related questions:
1. What does the similarity of triangles imply about their angles?
2. Can we use any specific triangle similarity criterion (e.g., AA, SAS) in this problem?
3. How do we know that corresponding sides in similar triangles are proportional?
4. Can we simplify the proportion method if we were solving for a single variable?
5. Is there another way to approach solving for \( x \) and \( y \) using geometric properties?

**Tip:** Always simplify and check the dimensions or relationships in diagrams like this one, especially when dealing with triangle similarity.

Find X and Y based on the geometry problem provided. The problem involves two triangles that appear to be similar, and the given expressions for the sides are: WZ = 2y + 15, ZY = 9y - 13, WX = x + 5, XY = 3x - 5.

Learn how to solve for X and Y in a geometry problem involving similar triangles. This problem covers key geometric concepts like triangle similarity and proportions, with expressions for the sides. Suitable for students in Grades 8-10.

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