https://file.aiquickdraw.com/mathbot/file/1726123996637e1gtwpd2.jpg

Let's analyze the information provided in the image to determine whether the triangles PQR and XYZ are congruent, similar, or unrelated for both cases.

### Case I:
- \(\angle P = 87^\circ\), \(\angle X = 87^\circ\)
- \(\ell(PQ) = 12\), \(\ell(XY) = 12\)
- \(\ell(PR) = 8\), \(\ell(XZ) = 18\)
- \(\ell(QR) = \text{not given}\), \(\ell(YZ) = \text{not given}\)

**Analysis**:
- The given angles \(\angle P\) and \(\angle X\) are equal.
- The sides \(\ell(PQ)\) and \(\ell(XY)\) are equal.

However, the second side pair \(\ell(PR)\) and \(\ell(XZ)\) are not proportional (\(\frac{8}{18} \neq \frac{12}{12}\)). Without the third side or more angles, the triangles are not similar or congruent based on the current information. Therefore, they are **unrelated**.

### Case II:
- \(\angle P = \angle X = \angle R = \angle Z = 90^\circ\)
- \(\ell(PQ) = 15\), \(\ell(XY) = 21\)
- \(\ell(PR) = 21\), \(\ell(XZ) = 24\)
- \(\ell(QR) = 24\), \(\ell(YZ) = 15\)

**Analysis**:
- All angles are equal (right angles in both triangles).
- Check if sides are proportional: 

\[
\frac{\ell(PQ)}{\ell(XY)} = \frac{15}{21} = \frac{5}{7}, \quad \frac{\ell(PR)}{\ell(XZ)} = \frac{21}{24} = \frac{7}{8}, \quad \frac{\ell(QR)}{\ell(YZ)} = \frac{24}{15} = \frac{8}{5}
\]

The ratios are not equal, so the triangles are not similar. Also, they are not congruent because the sides don't match exactly. Therefore, these triangles are also **unrelated**.

Would you like more details on these cases or have any questions?

### Related Questions:
1. What conditions are necessary for two triangles to be congruent?
2. What are the criteria for triangles to be considered similar?
3. Can triangles with the same angle but different side lengths be similar?
4. What is the difference between congruent and similar triangles?
5. How can you determine whether two triangles are unrelated using side lengths?

### Tip:
When analyzing triangles, always check if the given angles and sides satisfy any of the triangle congruence (SSS, SAS, ASA, AAS) or similarity (AA, SSS, SAS) criteria.

In each of the following cases, some information is given about the sides and/or angles of two triangles PQR and XYZ. From the given information, identify whether the two triangles PQR and XYZ are congruent, similar, or unrelated.

Analyze the sides and angles of triangles PQR and XYZ to determine whether they are congruent, similar, or unrelated. This problem involves applying triangle congruence and similarity theorems using SAS, SSS, and AA criteria. Suitable for students in Grades 9-12.

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