Answer :
To determine which statement is an example of the transitive property of congruence, let's review what the transitive property in mathematics entails, particularly in the context of congruence.
The transitive property states that if a first geometric figure is congruent to a second geometric figure, and the second geometric figure is congruent to a third geometric figure, then the first geometric figure is congruent to the third geometric figure.
Here’s a step-by-step analysis of each option:
Option A:
"If \( \triangle AEFG \) is congruent to \( \triangle AHJK \), then \( \triangle AHJK \) is congruent to \( \triangle AMNP \)."
This statement only describes congruence between two different pairs, but does not connect them through the transitive property. Therefore, it is not about the transitive property.
Option B:
"If \( \triangle AEFG \) is congruent to \( \triangle AHJK \), then \( \triangle AHJK \) is congruent to \( \triangle AEFG \)."
This statement employs the symmetric property of congruence, which states that if one figure is congruent to another, the second is congruent to the first. This is correct, but it's not an example of the transitive property.
Option C:
" \( \triangle AEFG \) is congruent to \( \triangle AEFG \)."
This statement expresses the reflexive property of congruence, where any geometric figure is congruent to itself. It does not illustrate transitivity.
Option D:
"If \( \triangle AEFG \) is congruent to \( \triangle AHJK \), and \( \triangle AHJK \) is congruent to \( \triangle AMNP \), then \( \triangle AEFG \) is congruent to \( \triangle AMNP \)."
This statement clearly exemplifies the transitive property of congruence. It shows that if the first triangle is congruent to the second, and the second is congruent to the third, then the first triangle is congruent to the third.
Therefore, the correct answer is:
D. If [tex]\( \triangle AEFG \)[/tex] is congruent to [tex]\( \triangle AHJK \)[/tex], and [tex]\( \triangle AHJK \)[/tex] is congruent to [tex]\( \triangle AMNP \)[/tex], then [tex]\( \triangle AEFG \)[/tex] is congruent to [tex]\( \triangle AMNP \)[/tex].
The transitive property states that if a first geometric figure is congruent to a second geometric figure, and the second geometric figure is congruent to a third geometric figure, then the first geometric figure is congruent to the third geometric figure.
Here’s a step-by-step analysis of each option:
Option A:
"If \( \triangle AEFG \) is congruent to \( \triangle AHJK \), then \( \triangle AHJK \) is congruent to \( \triangle AMNP \)."
This statement only describes congruence between two different pairs, but does not connect them through the transitive property. Therefore, it is not about the transitive property.
Option B:
"If \( \triangle AEFG \) is congruent to \( \triangle AHJK \), then \( \triangle AHJK \) is congruent to \( \triangle AEFG \)."
This statement employs the symmetric property of congruence, which states that if one figure is congruent to another, the second is congruent to the first. This is correct, but it's not an example of the transitive property.
Option C:
" \( \triangle AEFG \) is congruent to \( \triangle AEFG \)."
This statement expresses the reflexive property of congruence, where any geometric figure is congruent to itself. It does not illustrate transitivity.
Option D:
"If \( \triangle AEFG \) is congruent to \( \triangle AHJK \), and \( \triangle AHJK \) is congruent to \( \triangle AMNP \), then \( \triangle AEFG \) is congruent to \( \triangle AMNP \)."
This statement clearly exemplifies the transitive property of congruence. It shows that if the first triangle is congruent to the second, and the second is congruent to the third, then the first triangle is congruent to the third.
Therefore, the correct answer is:
D. If [tex]\( \triangle AEFG \)[/tex] is congruent to [tex]\( \triangle AHJK \)[/tex], and [tex]\( \triangle AHJK \)[/tex] is congruent to [tex]\( \triangle AMNP \)[/tex], then [tex]\( \triangle AEFG \)[/tex] is congruent to [tex]\( \triangle AMNP \)[/tex].
