Symmetric Property
The symmetric property is an essential property in algebra that is used in various math concepts such as equality, matrices, relations, congruence, etc. In general, the symmetric property on a set states that if one element is related to the second element, the second element is also related to the first element according to the relation defined on the set.
In this article, we will describe the symmetric property of equality, symmetric property of congruence, symmetric property of relations, and the symmetric property of matrices. We will solve various examples related to the symmetric property to better understand the concept.
1.  What is Symmetric Property? 
2.  Symmetric Property of Equality 
3.  Symmetric Property of Congruence 
4.  Symmetric Property of Matrix 
5.  Symmetric Property of Relations 
6.  FAQs on Symmetric Property 
What is Symmetric Property?
The symmetric property in algebra is defined as a property that implies if one element in a set is related to the other, then we can say that the second element is also related to the first element. We study different forms of symmetric properties as given below:
 Symmetric Property of Equality
 Symmetric Property of Congruence
 Symmetric Property of Relations
 Symmetric Property of Matrices
All symmetric properties are specific cases of the symmetric property of relations. For example, if we define a relation on the set of numbers as 'is equal to', then we get the symmetric property of equality. If the relation is defined on the set of geometric figures/triangles, then we get the symmetric property of congruence. Let us discuss the symmetric property of equality in the next section.
Symmetric Property of Equality
Now that we have understood the basic meaning of the symmetric property, let us describe the symmetric property of equality. It states that if a real number x is equal to a real number y, then we can say that y is equal to x. This property helps in proving the relation defined on the set of numbers as 'aRb if and only if a = b' to be symmetric relation. Mathematically, we can express the symmetric property of equality as 'If x = y, then y = x'. This property also helps in finding the value of variables in a system of equations.
Symmetric Property of Congruence
The symmetric property of congruence states that if a geometric figure is congruent to another, then we can say that the second figure is congruent to the first figure. For example, if triangle ABC is congruent to the triangle PQR, then we can also say that triangle PQR is congruent to the triangle ABC. Another example of the symmetric property of congruence is that if a line segment AB is congruent to another line segment CD, then we can say that CD is congruent to AB. We can use this property to angles, and other geometric figures.
Symmetric Property of Matrix
The symmetric property of a matrix states that if matrix A is symmetric, then matrix A is equal to its transpose, that is, A = A^{T}. Some of the important properties of a symmetric matrix are:
 Symmetric matrices are commutative, that is, if A and B are symmetric matrices, then AB = BA.
 The sum and difference of two symmetric matrices give the resultant a symmetric matrix.
 For integer n, if A is symmetric, ⇒ A^{n} is symmetric.
 If a matrix A is symmetric, then A^{1} is symmetric.
Symmetric Property of Relations
A relation R defined on a set A is said to be symmetric if for all a, b in A, if aRb then we must have bRa, that is, if (a, b) is in R, then (b, a) is in R. This symmetric property of relations is used to prove if a relation R is symmetric or an equivalence relation. The number of symmetric relations for a set having 'n' number of elements is given as N = 2^{n(n+1)/2}, where N is the number of symmetric relations and n is the number of elements in the set.
Important Notes on Symmetric Property
 Symmetric property of equality, symmetric property of congruence, symmetric property of relations, and the symmetric property of matrices are the different symmetric properties that we study in algebra.
 The symmetric property of equality states that if a real number x is equal to a real number y, then we can say that y is equal to x.
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Symmetric Property Examples

Example 1: If 3x + y = 4, then what is 4 equal to?
Solution: Using the symmetric property of equality, we have 3x + y = 4 ⇒ 4 = 3x + y
Answer: 4 is equal to 3x + y.

Example 2: If the line segment AB is congruent to the line segment CD, and AB = 5 cm, then find the length of CD.
Solution: Since the line segment AB is congruent to the line segment CD, therefore we have AB = CD and AB = 5 cm.
Using the symmetric property of congruence, we can write CD = AB = 5cm. So, CD = 5cm
Answer: The length of CD is 5 cm.

Example 3: If Mary's height is equal to that of Jane's, can we say that Jane's height is equal to that of Mary's?
Solution: Since Mary's height is equal to that of Jane's, we can write Mary = Jane. Now, using the symmetric property of equality, we can write Jane = Mary.
Answer: Yes, we can say that Jane's height is equal to that of Mary's.
FAQs on Symmetric Property
What is Symmetric Property in Geometry?
The symmetric property in geometry states that if one figure is congruent to another, then we can say that the second figure is congruent to the first figure. For example, if angle A is congruent to angle B, then we can say that angle B is congruent to angle B.
What is Symmetric Property of Equality?
The symmetric property of equality states that if a real number x is equal to a real number y, then we can say that y is equal to x. Mathematically, we can express the symmetric property of equality as 'If x = y, then y = x'.
What is Symmetric Property of Congruence?
The symmetric property of congruence states that if a geometric figure is congruent to another, then we can say that the second figure is congruent to the first figure.
How Do You Prove Symmetric Property?
We can prove the symmetric property of relation by assuming an element is related to another and then prove that the latter is also related to the former.
What is Symmetric Property of an Angle?
The symmetric property of an angle states that if angle A is congruent to angle B, then we can say that angle B is also congruent to angle A.
What is the Difference Between the Reflexive Property and Symmetric Property?
The reflexive property states that every element is related to itself and the symmetric property states that if one element is related to the second element, the second element is also related to the first element according to the relation defined on the set.
What is an Example of Symmetric Property?
Some of the important examples of symmetric properties are:
 If x + y = 3, then 3 = x + y
 If triangle ABC is congruent to triangle XYZ, then triangle XYZ is congruent to triangle ABC.
 If matrix A is symmetric, then it is equal to its transpose.
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