Question

If angle x is congruent to angle L, and P is congruent to angle M, angle A is congruent to angle K , line segment PA is congruent to line segment MK, line segment AX is congruent to line segment LL, and line segment XP is congruent to line segment LM, which option below shows a correct congruence statement?

Triangle PAX is congruent to Triangle MKL
Triangle XPA is congruent to Triangle MKL
Triangle XPA is congruent to Triangle KLM
Triangle PAX is congruent to Triangle KLM

Answer 1

To determine the correct congruence statement, we’ll summarize the given information and use it to check the relationships between the triangles.

1. Angles:
- \( \angle x \cong \angle L \)
- \( P \cong \angle M \)
- \( \angle A \cong \angle K \)

2. Line segments:
- \( PA \cong MK \)
- \( AX \cong LL \)
- \( XP \cong LM \)

Now, let's analyze the triangles formed:
- Triangle \( PAX \) has angles \( \angle P \), \( \angle A\), and \( \angle X \), with sides \( PA \), \( AX \), and \( XP \).
- Triangle \( MKL \) has angles \( \angle M \), \( \angle K\), and \( \angle L \), with sides \( MK \), \( KL \), and \( LM \).
- Triangle \( KLM \) consists of angles \( \angle K \), \( \angle L \), and \( \angle M \) and corresponds to sides \( KL \), \( LM \), and \( MK \).

Now, comparing angles and sides:
- From the information, we know:
- \( \angle P \cong \angle M \) (by \( P \cong M \))
- \( \angle A \cong \angle K \)
- \( \angle X \cong \angle L \)

And:
- \( PA \cong MK \)
- \( AX \cong LL \)
- \( XP \cong LM \)

Considering the triangle \( PAX \) and the triangle \( MKL \):
- The angles match as:
- \( \angle P \cong \angle M \)
- \( \angle A \cong \angle K \)
- \( \angle X \cong \angle L \)

And the sides:
- \( PA \cong MK \)
- \( AX \cong LL \)
- \( XP \cong LM \)

Thus, **Triangle \( PAX \) is congruent to Triangle \( MKL \)** using the Angle-Angle-Side (AAS) postulate.

Therefore, the correct congruence statement is:

**Triangle PAX is congruent to Triangle MKL**.