Congruency of Triangles Quick Check

Question

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

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

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

△PAX≅△MKL
triangle upper P upper A upper X 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

Answer 1

To determine the correct congruence statement based on the given information, we need to match the pairs of angles and sides correctly from triangles ∆XPA and ∆MKL.

Given the information:

• Angles: ∠X ≅ ∠L, ∠P ≅ ∠M, ∠A ≅ ∠K
• Sides: PA ≅ MK, AX ≅ KL, XP ≅ LM

Using the information provided, we can match corresponding parts of the triangles:

1. The angles correspond:

   • ∠X with ∠L
   • ∠P with ∠M
   • ∠A with ∠K

2. The sides correspond:

   • PA with MK
   • AX with KL
   • XP with LM

Based on the angle-side-angle (ASA) or side-angle-side (SAS) congruence criteria (since both angle measures and side lengths are provided), it’s appropriate to state that the triangles are congruent.

Now, let's look at the options:

• △XPA≅△MKL: This option corresponds correctly, matching angles and sides as indicated.
• △PAX≅△KLM: This mixes the order of vertices incorrectly; the triangles would not correspond properly.
• △PAX≅△MKL: Again, an incorrect order of vertices.
• △XPA≅△KLM: This mixes the triangles up incorrectly, as they should correspond to each other directly based on the given information.

Based on this analysis, the correct choice is:

△XPA≅△MKL.