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Determine whether the triangles are similar. If there is sufficient information to determine that the triangles are similar, write a similarity statement and explain your reasoning. And if there is not sufficient information, explain your reasoning.

 



To determine whether the triangles are similar, let's examine the given figure more closely. We have two right triangles, \( \triangle GHJ \) and \( \triangle KLJ \), sharing a common vertex \( J \) where the triangles intersect.

### Conditions for similarity
Triangles are similar if one of the following conditions is satisfied:
1. **Angle-Angle (AA) Similarity:** Two corresponding angles are congruent.
2. **Side-Angle-Side (SAS) Similarity:** Two sides are proportional, and the included angle is congruent.
3. **Side-Side-Side (SSS) Similarity:** All three sides of the triangles are proportional.

### Analyzing the figure:
- Both triangles \( \triangle GHJ \) and \( \triangle KLJ \) have right angles at \( \angle G \) and \( \angle K \), respectively.
- \( \angle HJ \) is a common angle to both triangles.

Thus, by the **Angle-Angle (AA) Similarity** criterion, the two triangles are similar because:
1. \( \angle G = \angle K = 90^\circ \) (corresponding right angles).
2. \( \angle HJ \) is common to both triangles.

### Conclusion:
Since two corresponding angles are congruent, the triangles \( \triangle GHJ \) and \( \triangle KLJ \) are **similar** by the AA criterion.

### Similarity Statement:
\[
\triangle GHJ \sim \triangle KLJ
\]

Would you like more details, or do you have any other questions?

Here are 5 questions to expand on this concept:
1. How do you apply the AA criterion for similarity in other types of triangles?
2. What is the significance of the shared angle \( \angle HJ \) in proving similarity?
3. Can we find the ratio of the sides in these similar triangles?
4. How does the right-angle property help in determining triangle similarity?
5. If additional side lengths were given, how would we use the SAS or SSS similarity criteria?

**Tip:** In geometric problems, always look for shared angles or parallel lines when trying to prove similarity between triangles!

Determine whether the triangles are similar. If there is sufficient information to determine that the triangles are similar, write a similarity statement and explain your reasoning. And if there is not sufficient information, explain your reasoning.

In this problem, we determine whether two triangles are similar by using the Angle-Angle (AA) similarity criterion. Both triangles share a right angle and a common angle, confirming their similarity.

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