If ∠X≅∠L, ∠P≅∠M, ∠A≅∠K, PA≅∠MK, AX≅∠KL, and XP≅∠LM, which option below shows a correct congruence statement?(1 point)

Question

Responses

A. △XPA≅△MKL
B. △XPA≅△KLM
C. △PAX≅△MKL
D. △PAX≅△KLM

Answer 1

To determine the correct congruence statement based on the given angle and side relationships, we can analyze the information provided:

1. Angles:

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

2. Sides:

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

To check the options, we'll use the Angle-Side-Angle (ASA) or Side-Angle-Side (SAS) congruence criteria:

1. For Option A: \(\triangle XPA \cong \triangle MKL\)

   • Angles: \(\angle X \cong \angle L\) and \(\angle P \cong \angle M\)
   • However, we do not have \(\angle A\) congruent to \(\angle K\). So this option does not work.

2. For Option B: \(\triangle XPA \cong \triangle KLM\)

   • Angles: \(\angle X \cong \angle K\) is not provided. So this option is invalid.

3. For Option C: \(\triangle PAX \cong \triangle MKL\)

   • Angles: \(\angle P \cong \angle M\) and \(\angle A \cong \angle K\)
   • Sides: \(PA \cong MK\). This option seems to fulfill the criteria.

4. For Option D: \(\triangle PAX \cong \triangle KLM\)

   • Same reasoning as above, and the congruences do not satisfy all conditions.

Since Option C (\(\triangle PAX \cong \triangle MKL\)) fits the congruence conditions based on given information, we conclude that this is the correct answer.

Answer: C. \(\triangle PAX \cong \triangle MKL\)