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To determine if the triangles in parts (a) and (b) are similar, let's analyze each pair using the criteria for triangle similarity, which include:

1. **Angle-Angle (AA) Similarity**: Two triangles are similar if two angles in one triangle are equal to two angles in the other triangle.
2. **Side-Angle-Side (SAS) Similarity**: Two triangles are similar if two sides in one triangle are proportional to two sides in the other triangle, and the included angles are equal.
3. **Side-Side-Side (SSS) Similarity**: Two triangles are similar if all three pairs of corresponding sides are proportional.

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### Part (a)

**Given information**:
- **Triangle ABC**: \( \angle ABC = 112^\circ \), \( \angle ACB = 41^\circ \), and side \( BC = 8 \, \text{cm} \).
- **Triangle XYZ**: \( \angle XYZ = 112^\circ \), \( \angle YZX = 41^\circ \), and side \( ZX = 9 \, \text{cm} \).

**Analysis**:
- Both triangles have two angles that match (\(112^\circ\) and \(41^\circ\)), which means by the **Angle-Angle (AA) Similarity** criterion, they are similar.

**Conclusion**:
Yes, \( \triangle ABC \sim \triangle XYZ \) by the AA Similarity criterion.

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### Part (b)

**Given information**:
- **Triangle JKL**: \( \angle KJL = 95^\circ \), side \( JK = 12 \, \text{m} \), and side \( JL = 15 \, \text{m} \).
- **Triangle MNO**: \( \angle NMO = 95^\circ \), side \( MN = 4 \, \text{m} \), and side \( MO = 5 \, \text{m} \).

**Analysis**:
- Both triangles have an angle of \(95^\circ\), so if the sides around this angle are proportional, the triangles will be similar by the **Side-Angle-Side (SAS) Similarity** criterion.
- Calculating the side ratios:
  \[
  \frac{JK}{MN} = \frac{12}{4} = 3, \quad \frac{JL}{MO} = \frac{15}{5} = 3
  \]
- Since both pairs of corresponding sides are proportional, and the included angle is equal, the triangles are similar by the SAS criterion.

**Conclusion**:
Yes, \( \triangle JKL \sim \triangle MNO \) by the SAS Similarity criterion.

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Would you like further explanation on any part of this solution, or do you have any specific questions?

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### Follow-up Questions

1. What other similarity criteria are there beyond AA, SAS, and SSS?
2. How can you determine the proportionality ratio between two similar triangles?
3. Are there cases where triangles can look similar but do not meet similarity criteria?
4. How does the AA similarity criterion help in proving similarity without knowing side lengths?
5. How could similarity be used to calculate unknown sides in real-world problems?

### Tip

When working with triangles, remember that if two angles match, the third angle must also match (since the sum of angles in a triangle is always 180°). This is why the AA criterion is often enough to prove similarity.

Are the following triangles similar? Explain why or why not. If similar, name the property of similarity of triangles to justify your answer.

Determine if two triangles are similar using the Angle-Angle (AA) and Side-Angle-Side (SAS) similarity criteria. Analyze given angles and side lengths to confirm triangle similarity with detailed explanations.

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