https://file.aiquickdraw.com/mathbot/file/1730232032631l45fig4r.jpg

To determine if triangles \( \triangle ABC \) and \( \triangle ADB \) are similar, we need to see if they meet one of the similarity criteria:

1. **Angle-Angle (AA) Similarity**: Two triangles are similar if two pairs of corresponding angles are equal.
2. **Side-Angle-Side (SAS) Similarity**: Two triangles are similar if one angle is equal in both triangles and the sides including these angles are proportional.
3. **Side-Side-Side (SSS) Similarity**: Two triangles are similar if all corresponding sides are proportional.

### Analysis of the Given Angles:
In the image:
- \( \angle ABC = 39^\circ \)
- \( \angle ACB = 39^\circ \)

Since \( \triangle ABC \) is a triangle, we know that the sum of its internal angles is \(180^\circ\):
\[
\angle BAC = 180^\circ - 39^\circ - 39^\circ = 102^\circ.
\]

For \( \triangle ADB \):
- \( \angle ADB \) also shares the angle \( \angle ABC = 39^\circ \).
- \( \triangle ADB \) shares the same \( \angle BAC = 102^\circ \) with \( \triangle ABC \).

### Conclusion:
Since \( \triangle ABC \) and \( \triangle ADB \) have two pairs of corresponding angles that are equal (\( \angle ABC = 39^\circ \) and \( \angle BAC = 102^\circ \)), they are similar by the **Angle-Angle (AA) Similarity Criterion**.

Would you like further clarification on this, or have any other questions?

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Here are 5 related questions to explore further:
1. How would the similarity of these triangles change if one angle was different?
2. What other methods can we use to prove similarity besides AA?
3. How does the proportionality of sides relate to similar triangles?
4. Can we determine if the triangles are congruent based on this information?
5. How do similarity criteria apply to other polygon types, like quadrilaterals?

**Tip:** Remember that two triangles with two equal angles are always similar, as the third angle will automatically be the same due to the angle sum property.

Are triangles ABC and ADB similar? How do you know?

Explore how to determine if triangles ABC and ADB are similar using the Angle-Angle (AA) Similarity Criterion. Understand the steps to prove similarity by identifying corresponding angles. Suitable for students in Grades 7-9.

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