Question

congruency of triangles quick check
if \\( \angle x \cong \angle l, \angle p \cong \angle m, \angle a \cong \angle k, \overline{p a} \cong \overline{m k}, \overline{a x} \cong \overline{k l} \\), and \\( \overline{x p} \cong \overline{l m} \\), which option below shows a correct congruence statement?
(1 point)
\\( \triangle x p a \cong \triangle k l m \\)
\\( \triangle p a x \cong \triangle m k l \\)
\\( \triangle x p a \cong \triangle m k l \\)
\\( \triangle p a x \cong \triangle k l m \\)

Explanation:

Step1: Match the corresponding vertices

Given \(\angle X\cong\angle L\), \(\angle P\cong\angle M\), \(\angle A\cong\angle K\), \(\overline{PA}\cong\overline{MK}\), \(\overline{AX}\cong\overline{KL}\), \(\overline{XP}\cong\overline{LM}\).
The order of vertices in congruent triangles should match the congruent angles and sides.

Step2: Analyze each option

• For \(\triangle XPA\cong\triangle KLM\): The correspondence of angles and sides is not correct.
• For \(\triangle PAX\cong\triangle MKL\):
• \(\angle P\) corresponds to \(\angle M\), \(\angle A\) corresponds to \(\angle K\), \(\angle X\) corresponds to \(\angle L\)
• \(\overline{PA}\cong\overline{MK}\), \(\overline{AX}\cong\overline{KL}\), \(\overline{XP}\cong\overline{LM}\)
• For \(\triangle XPA\cong\triangle MKL\): The correspondence of angles and sides is not correct.
• For \(\triangle PAX\cong\triangle KLM\): The correspondence of angles and sides is not correct.

Answer:

\(\triangle PAX\cong\triangle MKL\)