Which statement is an example of the symmetric property of congruence?
A. If AKLM ≅ PQR, then PQR ≅ AKLM.
B. If AKLM ≅ PQR, then PQR ≅ STU.
C. AKLM ≅ AKLM
D. If AKLM ≅ PQR, and PQR ≅ STU, then AKLM ≅ STU.
Answer (1)
The symmetric property of congruence states that if A ≅ B , then B ≅ A .
Option A: If A K L M ≅ A PQR , then A PQR ≅ A K L M matches the definition.
Options B, C, and D represent the transitive and reflexive properties, respectively.
Therefore, the correct answer is A: I f A K L M ≅ A PQR , t h e n A PQR ≅ A K L M .
Explanation
Understanding the Symmetric Property The symmetric property of congruence states that if one geometric figure is congruent to another, then the second geometric figure is congruent to the first. In mathematical terms, if A ≅ B , then B ≅ A . We need to identify the statement that correctly represents this property.
Analyzing the Options Let's analyze each option:
Option A: If A K L M ≅ A PQR , then A PQR ≅ A K L M . This statement perfectly matches the definition of the symmetric property of congruence.
Option B: If A K L M ≅ A PQR , then A PQR ≅ A ST U . This statement does not represent the symmetric property. It suggests a relationship between three figures, which is more akin to the transitive property.
Option C: A K L M ≅ A K L M . This statement shows that a figure is congruent to itself, which is the reflexive property, not the symmetric property.
Option D: If A K L M ≅ A PQR , and A PQR ≅ A ST U , then A K L M ≅ A ST U . This statement describes the transitive property of congruence, where if the first figure is congruent to the second, and the second is congruent to the third, then the first is congruent to the third.
Identifying the Correct Statement Based on our analysis, option A is the correct representation of the symmetric property of congruence.
Examples
The symmetric property is useful in various geometric proofs. For example, if you're proving that two triangles are congruent and you've shown that triangle ABC is congruent to triangle DEF, the symmetric property allows you to immediately state that triangle DEF is congruent to triangle ABC, which can be helpful in further steps of your proof. This property ensures that congruence is a reversible relationship, simplifying logical deductions in geometry.
