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!