https://file.aiquickdraw.com/mathbot/file/1726038775014hqgcxuek.jpg

The image shows a series of geometry problems focused on similar triangles and quadrilaterals, where the task is to find the value of unknowns marked with variables.

Here’s a breakdown of the problems based on the image:

1. **(a) ΔABC is similar to ΔDEF**:
   - Triangles have corresponding angles that are equal. The known side lengths are:
     - ΔABC: \( AB = 12 \), \( BC = 18 \), and \( ∠B = 30^\circ \)
     - ΔDEF: \( DE = x \), \( EF = y \)
   - Use the similarity ratio to find \(x\) and \(y\).

2. **(b) ΔABC is similar to ΔVWZ**:
   - The given sides are:
     - ΔABC: \( AB = 12 \), \( BC = 18 \), \( AC = z \)
     - ΔVWZ: \( VW = 20 \), \( WZ = 24 \)
   - Apply the properties of similar triangles to find \(z\).

3. **(c) ABCD is similar to EFGH**:
   - Quadrilaterals ABCD and EFGH are given as similar:
     - ABCD: \( AB = 9 \), \( BC = 15 \), \( ∠A = 75^\circ \)
     - EFGH: \( EF = x \), \( FG = 10 \), \( ∠E = 75^\circ \)
   - Solve for \(x\) using similarity.

4. **(d) ABCD is similar to PQRS**:
   - Given information:
     - ABCD: \( AB = 12 \), \( BC = x \), \( CD = 15 \)
     - PQRS: \( PQ = 9 \), \( QR = y \), \( RS = 6 \)
   - Use the similarity ratio to find both \(x\) and \(y\).

Would you like a step-by-step solution for one of these problems, or all of them? Let me know if you need more details.

### Five related questions:
1. How do you find the similarity ratio between two triangles?
2. Why are the corresponding angles of similar triangles always equal?
3. How can you use proportions to find unknown side lengths in similar figures?
4. What is the significance of angle congruence in establishing similarity between quadrilaterals?
5. How can you verify if two figures are similar using side and angle information?

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

In each of the following, find the value of each marked unknown. The dimensions given are in centimeters. (a) ΔABC is similar to ΔDEF, (b) ΔABC is similar to ΔVWZ, (c) ABCD is similar to EFGH, (d) ABCD is similar to PQRS.

Learn how to solve problems involving similar triangles and quadrilaterals. This step-by-step guide covers finding unknown side lengths using similarity ratios, as well as applying theorems such as AA Similarity and Proportionality in triangles and quadrilaterals. Suitable for students in Grades 8-10.

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