Name the property of congruence that justifies this statement if ∠a= ∠b and ∠b = ∠c then ∠a = ∠c
A: transitive property
B: symmetric property
C:reflexive property
D:none of these
B: symmetric property
The property of congruence that justifies the statement "if ∠a= ∠b and ∠b = ∠c, then ∠a = ∠c" is:
B: symmetric property.
The symmetric property of congruence states that if two angles are congruent, then switching the order of the angles will still result in congruence. In this case, since ∠a = ∠b and ∠b = ∠c, the symmetric property allows us to conclude that ∠a = ∠c.
The property of congruence that justifies the given statement is the transitive property of congruence.
To understand why, let's look at the definitions of these properties:
1. Reflexive property: This property states that an angle or shape is congruent to itself. For example, if ∠a is congruent to ∠a, we can say that ∠a = ∠a. However, this property does not help us prove that ∠a = ∠c based on ∠a = ∠b and ∠b = ∠c.
2. Symmetric property: This property states that if ∠a is congruent to ∠b, then ∠b is also congruent to ∠a. While this property allows us to switch the order of congruence (e.g., if ∠a = ∠b, then ∠b = ∠a), it doesn't help in proving that ∠a = ∠c based on ∠a = ∠b and ∠b = ∠c.
3. Transitive property: This property states that if ∠a is congruent to ∠b and ∠b is congruent to ∠c, then ∠a is congruent to ∠c. In other words, if two angles are congruent to a third angle, then they must also be congruent to each other. This property allows us to conclude that ∠a = ∠c based on the given information (∠a = ∠b and ∠b = ∠c).
Therefore, the correct answer is A: transitive property.