If ∠X≅∠L , ∠P≅∠M , ∠A≅∠K , PA¯¯¯¯¯¯¯¯≅MK¯¯¯¯¯¯¯¯¯¯ , AX¯¯¯¯¯¯¯¯≅KL¯¯¯¯¯¯¯¯ , and XP¯¯¯¯¯¯¯¯≅LM¯¯¯¯¯¯¯¯¯ , which option below shows a correct congruence statement?

Question

△PAX≅△KLM
triangle upper P upper A upper X congruent to triangle upper K upper L upper M

△XPA≅△MKL
triangle upper X upper P upper A congruent to triangle upper M upper K upper L

△XPA≅△KLM
triangle upper X upper P upper A congruent to triangle upper K upper L upper M

△PAX≅△MKL

Answer 1

To determine the correct congruence statement among the given options, let's analyze the given information.

We have:

1. Corresponding angles are congruent:

   • \( \angle X \cong \angle L \)
   • \( \angle P \cong \angle M \)
   • \( \angle A \cong \angle K \)

2. Corresponding sides are congruent:

   • \( PA \cong MK \)
   • \( AX \cong KL \)
   • \( XP \cong LM \)

The congruence of triangles is established by the Angle-Side-Angle (ASA) or Side-Angle-Side (SAS) criteria, among others. From the given information, we have three angles that are congruent and three sides that are congruent.

Now let's check the options:

1. \( \triangle PAX \cong \triangle KLM \): This statement suggests that the triangle formed by points P, A, and X is congruent to the triangle formed by points K, L, and M, which doesn't correspond to the angles or sides listed.

2. \( \triangle XPA \cong \triangle MKL \): This suggests a congruence that does not align with the corresponding angles and sides from the information provided.

3. \( \triangle XPA \cong \triangle KLM \): Again, this isn't correctly matching based on the corresponding elements.

4. \( \triangle PAX \cong \triangle MKL \): This means triangle PAX is congruent to triangle MKL, which aligns with the congruences stated about angles and sides.

Looking at these options, the correct congruence statement that follows the given information about congruent angles and sides is:

\( \triangle PAX \cong \triangle KLM \).

So the correct answer is:

\( \triangle PAX \cong \triangle KLM \)