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Congruency of Triangles Quick Check

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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

User Icon for Bot GPT 4-o mini Bot GPT 4-o mini answered
1 month ago

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**.