https://file.aiquickdraw.com/mathbot/file/1727380894201yaxz740j.jpg

The given figure consists of two triangles: \(\triangle MNL\) and \(\triangle QPR\), with the task asking to identify the similarity criterion between these two triangles.

### Step-by-step reasoning:

1. **Identifying Corresponding Angles**:
   - Both triangles have an angle of \(70^\circ\), i.e., \(\angle MNL = \angle QPR = 70^\circ\).
   
2. **Comparing Side Ratios**:
   - In \(\triangle MNL\), the sides are \(MN = 2.5\), \(ML = 5\), and \(NL = 5\).
   - In \(\triangle QPR\), the sides are \(PQ = 5\), \(QR = 10\), and \(PR = 5\).
   
   We can calculate the ratios of corresponding sides:
   - \(\frac{MN}{PQ} = \frac{2.5}{5} = 0.5\)
   - \(\frac{ML}{PR} = \frac{5}{5} = 1\)
   - \(\frac{NL}{QR} = \frac{5}{10} = 0.5\)

3. **Checking for Similarity Criterion**:
   - The two triangles share a common angle of \(70^\circ\), and the ratios of corresponding sides are not equal (i.e., \(0.5\), \(1\), \(0.5\)).
   - Therefore, the triangles cannot be similar by **SSS (Side-Side-Side)** or **SAS (Side-Angle-Side)** criterion since the side ratios do not match uniformly.

   However, since they share an equal angle and have proportional corresponding sides in at least two cases (with a scaling factor between the triangles), the **Angle-Angle (AA)** similarity criterion applies.

### Conclusion:
The similarity criterion used here is **Angle-Angle (AA) Similarity** because both triangles share the same angle of \(70^\circ\), and at least two corresponding sides are proportional.

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Do you want further details or have any questions?

Here are some related questions to expand on this:
1. How do you calculate the ratio of corresponding sides for similar triangles?
2. What is the difference between AA similarity and SSS similarity?
3. Can two triangles be similar if only one pair of angles is congruent?
4. How can the similarity criterion help in solving real-world problems?
5. What are the conditions for two triangles to be congruent instead of similar?

**Tip:** Always check if triangles share proportional sides and corresponding angles to determine similarity.

In the given figure, ΔMNL ∼ ΔQPR. Name the similarity criterion used here.

In this geometry problem, learn how to apply the Angle-Angle (AA) similarity criterion to determine the similarity between triangles MNL and QPR. We analyze corresponding angles and side ratios to confirm triangle similarity. Suitable for students in Grades 8-10.

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