Answer :
To determine which statement exemplifies the symmetric property of congruence, we first need to understand what the symmetric property of congruence means in geometry. The symmetric property of congruence states that if one geometrical figure is congruent to another, then the second figure is also congruent to the first. Mathematically, if [tex]\( A \cong B \)[/tex], then [tex]\( B \cong A \)[/tex].
Let's analyze each option:
Option A: [tex]\( AEFG \cong AEFG \)[/tex]
This option states that figure AEFG is congruent to itself. This is an example of the reflexive property, not the symmetric property. The reflexive property states that any figure is congruent to itself.
Option B: If [tex]\( AEFG \cong AHJK \)[/tex], and [tex]\( AHJK \cong AMNP \)[/tex], then [tex]\( AEFG \cong AMNP \)[/tex].
This statement talks about transitivity. The transitive property states that if one figure is congruent to a second and the second is congruent to a third, then the first figure is congruent to the third. This is not the symmetric property.
Option C: If [tex]\( AEFG \cong AHJK \)[/tex], then [tex]\( AHJK \cong AEFG \)[/tex].
This statement correctly aligns with the symmetric property of congruence. It illustrates that if figure AEFG is congruent to figure AHJK, then figure AHJK is also congruent to figure AEFG.
Option D: If [tex]\( AEFG \cong AHJK \)[/tex], then [tex]\( AHJK \cong AMNP \)[/tex].
This statement does not represent any known property of congruence because it suggests a direct relationship between figures that are not compared initially. It does not show symmetry or reflexivity, nor does it follow the transitive relationships correctly.
Therefore, the correct statement that exemplifies the symmetric property of congruence is:
C. If AEFG ≅ AHJK, then AHJK ≅ AEFG.
Let's analyze each option:
Option A: [tex]\( AEFG \cong AEFG \)[/tex]
This option states that figure AEFG is congruent to itself. This is an example of the reflexive property, not the symmetric property. The reflexive property states that any figure is congruent to itself.
Option B: If [tex]\( AEFG \cong AHJK \)[/tex], and [tex]\( AHJK \cong AMNP \)[/tex], then [tex]\( AEFG \cong AMNP \)[/tex].
This statement talks about transitivity. The transitive property states that if one figure is congruent to a second and the second is congruent to a third, then the first figure is congruent to the third. This is not the symmetric property.
Option C: If [tex]\( AEFG \cong AHJK \)[/tex], then [tex]\( AHJK \cong AEFG \)[/tex].
This statement correctly aligns with the symmetric property of congruence. It illustrates that if figure AEFG is congruent to figure AHJK, then figure AHJK is also congruent to figure AEFG.
Option D: If [tex]\( AEFG \cong AHJK \)[/tex], then [tex]\( AHJK \cong AMNP \)[/tex].
This statement does not represent any known property of congruence because it suggests a direct relationship between figures that are not compared initially. It does not show symmetry or reflexivity, nor does it follow the transitive relationships correctly.
Therefore, the correct statement that exemplifies the symmetric property of congruence is:
C. If AEFG ≅ AHJK, then AHJK ≅ AEFG.
