To determine if the Hypotenuse-Leg (HL) Congruence Property can be applied, let's first review the conditions for HL Congruence:

• The HL theorem applies only to right triangles.
• It states that if two right triangles have a congruent hypotenuse and one congruent leg, then the triangles are congruent.

Now, let's examine each pair of triangles in the image:

(a)

• Both triangles are right triangles (shown by the right angle symbol).
• The hypotenuses and one leg of both triangles are marked as congruent (20 km for the hypotenuse and 12 km for the leg).
• Conclusion: Yes, HL Congruence Property can be used.

(b)

• These triangles have right angles and a congruent leg (marked on the figure), but the hypotenuses are not marked as congruent.
• Conclusion: No, HL Congruence Property cannot be used because the hypotenuses are not shown to be congruent.

(c)

• These triangles do not have right angles (they are not right triangles).
• Conclusion: No, HL Congruence Property cannot be used because the triangles are not right triangles.

(d)

• The triangles are right triangles (indicated by the right angle symbols).
• One leg is congruent (marked on the figure), but the hypotenuses are not marked as congruent.
• Conclusion: No, HL Congruence Property cannot be used because the hypotenuses are not shown to be congruent.

Summary:

• (a) Yes
• (b) No
• (c) No
• (d) No

Would you like further explanation on any part of this?

Here are some related questions to expand your understanding:

1. What is the difference between the HL theorem and the SSS (Side-Side-Side) theorem?
2. Can HL be applied if only the hypotenuse is congruent, but no legs are congruent?
3. How does the Pythagorean Theorem relate to the HL Congruence Property?
4. In what other scenarios can we prove two triangles are congruent besides HL?
5. Why do we only apply the HL Congruence Property to right triangles?

Tip: Remember, the HL Congruence Property can only be applied to right triangles, so always check for the right angle first!