Imagine you and your best friend are wearing matching shirts. If your friend is wearing the same shirt as you, then we can also say that you are wearing the same shirt as them, right?
In math, we would express this as: if A = B, then B = A. Here, your friend’s shirt is A and your shirt is B.
We call this simple and important math rule the symmetric property.
In this guide, we’ll explore what the symmetric property is, how it works in math, and why it’s important in arithmetic, algebra, geometry, and real life.
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The symmetric property is a math rule that says that if one value is equal to another, then the second value must also be equal to the first.
In math terms, if A = B, then B = A. For example:
If 5 + 3 = 8, then 8 = 5 + 3.
If x = 12, then 12 = x.
At first, the symmetric property might seem like a simple and obvious rule, but it plays a big role in math. It helps us use logic to understand how numbers, shapes, and equations connect.
It’s part of the axioms – or, basic truths – that mathematicians use to build more complex math concepts. Without it, math wouldn’t work as smoothly, and solving problems would be way more confusing.
The symmetric property is one of the basic rules of math, called laws of equality. Ancient Greek mathematicians were some of the first to write down and use this idea, and over time, it became an important part of algebra.
Today, we use it all the time when solving equations and proving math rules.
One of the biggest reasons this property matters is that it keeps math fair and predictable. If one thing is equal to another, flipping it around shouldn’t change anything.
Imagine you're in a video game and you pick up a magic potion that gives you +10 health points. The game tells you "Potion = +10 HP." If you check your status, it will also say "+10 HP = Potion." No matter which way you write it, the potion still gives you the same health boost.
Or think about a seesaw at the park. If you and a friend weigh the same, the seesaw stays balanced. Now, imagine switching seats—nothing changes! The seesaw stays level, just like an equation when the symmetric property is applied.
Just like a balanced seesaw stays level, the symmetric property shows that if two values are equal, their positions can be switched without changing the equation.
The symmetric property helps us understand that equality works both ways.
Ancient Greek mathematicians were some of the first to write down and use the idea of the symmetric property
The symmetric property shows up in math and in everyday life, and you’ve probably used it without even realizing it.
In simple arithmetic, if one number equals a combination of other numbers, then you can reverse the order, and it will still be true.
If 10 = 7 + 3, then we can also say 7 + 3 = 10.
No matter which way we write it, the sum remains the same.
Symmetric property shows that equality works in both directions.
The symmetric property is useful when solving equations. It allows us to rewrite expressions in a way that it’s easier to solve for variables.
A simple example would be, if x = 4, then we can also write 4 = x.
At first, this might not seem that important, but when solving algebraic equations, being able to flip an equation can help us organize our work and check our answers easier.
We used the symmetric property in a recent guide to derive the point-slope form of a line and turn it into a format that is easier to work with.
Deriving the point-slope form, we arrived at: \( \displaystyle m(x + x_1) = y - y_1 \).
Notice how the terms on the left are more complex than those on the right? To make further operations with the point form easier, such as isolating the y, we used the symmetric property to swap the sides and arrived at: \( \displaystyle y - y_1 = m(x + x_1) \)
We also use the symmetric property in geometry when working with congruent angles and shapes.
If two angles or figures are congruent, it does not matter which one is written first—they will always be equal to each other.
In math, we say: If angle A ≅ angle B, then angle B ≅ angle A.
Or, if two triangles are congruent, all their corresponding sides and angles match, no matter how you name them.
This is useful when proving geometric theorems and solving problems involving triangles, circles, and other shapes.
The symmetric property also applies to real-life situations where things are equal or balanced.
We used the shirt example where if you can say that your shirt is the same as your friend’s, then your friend can say that their shirt is the same as yours.
Another real-life example is trading objects.
If you trade your friend a pencil for a pen, then they could also say they traded a pen for a pencil. The trade goes both ways, just like an equation using the symmetric property.
The symmetric property helps explain fairness and balance in everyday situations.
There are four main types of symmetric properties in math.
In middle school math, the symmetric property of equality helps you solve algebra equations.
As you move into geometry, the symmetric property of congruence will help you understand shapes, angles, and proofs.
And even though you might not use the symmetric property of matrices yet, it is exciting to know that symmetry plays a role in science, technology, and the real world.
The symmetric property of equality is the most common type of symmetric property and the one we use the most in algebra.
We have explored this type already: If two things are equal, we can swap them around and they’ll still be equal.
If we know that 7 = 3 + 4, then flipping it around to 3 + 4 = 7 doesn’t change anything—it’s still true!
The symmetric property of congruence applies to geometry and the study of shapes and it states that if two shapes or angles are congruent, which means they are identical in size and shape, then it does not matter which one you list first—they will still be equal.
If ΔABC ≃ ΔXYZ, then ΔXYZ ≃ ΔABC. It doesn’t matter which triangle we write first.
If angle A ≃ angle B, then angle B ≃ angle A, because congruent angles are equal no matter how we order them.
This rule is important when proving geometric theorems and working with reflections, rotations, and translations.
If two figures are congruent, we can flip them, turn them, or slide them around, and they will still be exactly the same. Imagine cutting out two identical shapes from a piece of paper. If you switch them around, they will still be the same shape.
Relationships aren’t just about math—they happen between people, places, and objects too! The symmetric property of relations tells us that some relationships go both ways, but not all of them.
Let’s start with one that is symmetric.
If Lily is friends with Jake, then Jake is also friends with Lily. That’s how friendship works—if it’s mutual, it goes both ways.
Or imagine two cities connected by a two-way street—if City A is connected to City B, then City B is also connected to City A.
But not all relationships work like that.
For example, if Lily is Jake’s older sister, that does not mean Jake is Lily’s older sister. That would be pretty confusing, right? That’s an example of a relationship that is not symmetric.
Some relationships, like friendships, trading, and high-fives, are symmetric – they go both ways. Others, like winning and losing a game, or being someone’s parent, are not symmetric because they only work in one direction.
The symmetric property of relations helps us understand real-world connections, and more specifically, it is used in logic, graph theory, and data organization.
One of the properties that Mathnasium students often confuse with the symmetric property is the commutative property. While they might look alike, they are actually very different!
The commutative property is about changing the order of numbers in an addition or multiplication problem.
It tells us that switching numbers around does not change the answer, like so:
\( \displaystyle 2 + 3 = 3 + 2 \)
\( \displaystyle 4 \times 5 = 5 \times 4 \)
No matter how you arrange the numbers, you still get the same result.
The symmetric property, on the other hand, is about flipping an equation across the equal sign:
If 5 = 2 + 3, then 2 + 3 = 5.
If x = 10, then 10 = x.
A simple way to remember the difference between the two is that commutative property changes order, while symmetric property flips the equation.
Think of them as math cousins. The commutative property works inside the numbers on one side of the equal sign, while the symmetric property focuses on the equal sign itself.
The symmetric property might seem like common sense, but it is actually an important rule that keeps math logical and consistent. Without it, we wouldn’t be able to rearrange equations, which is a key part of solving algebra problems.
If you solve x = 9, you might not think twice about writing 9 = x, but that’s the symmetric property in action.
Even simple rules like this help build the foundation for more advanced math, including geometry, algebra, and calculus.
No, the symmetric property only works for addition and multiplication, not subtraction or division. This is because switching the order in subtraction and division can change the result. For example:
8 - 5 ≠ 5 - 8 (8 minus 5 is 3, but 5 minus 8 is -3)
12 ÷ 4 ≠ 4 ÷ 12 (12 divided by 4 is 3, but 4 divided by 12 is \( \displaystyle \frac{1}{3} \))
Since the answers are not the same, the symmetric property does not apply to subtraction and division.
No, the symmetric property only works with an equal sign (=) because equality means both sides are the same. Inequalities (<, >, ≤, ≥) do not always work both ways.
If 3 < 5, does that mean 5 < 3? No! 3 is smaller than 5, but 5 is not smaller than 3.
Since flipping inequalities can change their meaning, the symmetric property does not apply to inequalities.
Yes! The symmetric property works with all real numbers, including negative numbers and fractions. As long as there is an equal sign (=), you can flip the equation, and it will still be true.
If -7 = -3 - 4, then -3 - 4 = -7.
If 1/2 = 3/6, then 3/6 = 1/2.
This works because negative numbers and fractions still follow the same rules of equality as whole numbers. The order in which they are written does not change the fact that they are equal.
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