102 LearnersLast updated on July 5th, 2025
Transitive Property
The transitive property states that if a number p is equal to q and q is equal to r, then p is equal to r. In other words, the transitive property states that, p = q, q = r, then p = r. This property is also known as the transitive property of equality. This article explores the transitive property of equality, inequality, congruence, and its uses.
What is Transitive Property?
In mathematics, the transitive property is one of the properties of equality. The word “transitive” means passing a relation from one element to another through a common intermediary. The transitive property states that if two numbers are related to each other by some rule, and if the second number is related to the third number, then the first number is related to the third number. If a = b and b = c, then a = c. For example, if the number of pens is the same as the number of pencils, and the number of pencils is the same as the number of books. Then according to the transitive property, the number of pens is the same as the number of books.
What is the Transitive Property of Equality?
The transitive property states if a = b and b = c, then a = c. This means for three quantities, a, b, and c where if ‘a’ is related to b and b is related to c in the same way, then ‘a’ is related to ‘c’. The two properties of equality related to the transitive property are:
- Reflexive Property of Equality: According to the reflexive property of equality, any real number is equal to itself. That is, for any real number ‘a’, the reflexive property states that a = a.
- Symmetric Property of Equality: The symmetric property states that if a quantity is equal to another, the order of equality does not affect the result. That is if a = b, then b = a.
What is the Transitive Property of Inequality?
The transitive property applies for both equality and inequality. The transitive property of inequality states that if a ≤ b and b ≤ c, then a ≤ c. If the first number is less than or equal to the second number, and the second number is less than or equal to the third number.
For example, if a ≤ 3 and 3 ≤ b, then a ≤ b, similarly, 5 ≤ 7 and 7 ≤ 8, therefore 5 ≤ 8.
What is the Transitive Property of Congruence?
The transitive property of congruence states that if two shapes are congruent to the third shape, then they are congruent to each other. For example, for △ABC, △PQR, and △STU, if △ABC is congruent to △PQR and △PQR is congruent to △STU. The transitive property of congruence states that △ABC is congruent to △STU.
How to Use Transitive Property?
The transitive property is used in the fields of math, including algebra, geometry, arithmetic, etc. In geometry, the transitive property is used to analyze the equal or congruent quantities which follow the same rule.
Transitive property of angle: The transitive property of congruence (angle) states that if ∠M ≅ ∠N and ∠N ≅ ∠O, then ∠M ≅ ∠O
Transitive property of parallel lines: According to the transitive property of parallel lines, if line p is parallel to line q, and line q is parallel to line r, then line p is parallel to line r.
Transitive property of inequality of real numbers: The transitive property is applicable for both equality and inequality, as discussed. The transitive property of inequality of real numbers states that if a < b, and b < c, then a < c. It is even applicable in reverse order, that is if a > b and b > c, then a > c.
Construction of equilateral triangles using the transitive property:
To draw an equilateral triangle using the transitive property, follow these steps.
Draw segment AB and two circles with radius, AB. The point of intersection can be labeled as point C. Join the points A, B, and C to form a triangle.
For the circle with center A, the radii of the circle are AB and AC. This means AB = AC
The radii of the circle with center C are AC and BC, so AC = BC
Since, AB = AC and AC = BC, by transitive property, AB = BC
Therefore, the triangle ABC is equilateral, as the three sides of the triangle are equal.
Common Mistakes and How to Avoid Them in Transitive Property
For solving equations, inequalities, geometry, and so on, understanding the transitive property is important. Students usually tend to make mistakes and misapply the property. In this section, let’s learn some common mistakes and the ways to avoid them to master transitive property.
Mistake 1
Mixing up the transitive and symmetric properties
Students confuse transitive with the symmetric properties; it can lead to errors in reasoning. In transitive property, if a=b and b=c, then a=c. While, the symmetric property states that if a = b, then b = a.
Mistake 2
Incorrectly using the transitive property in inequalities
When the inequalities point in the same direction, then transitive property is applicable. For example, if a < b and b < c, then a < c, or if a > b and b > c, then a > c.
Mistake 3
Confusing congruence with equality
Students often confuse and think that ∠A ≅ ∠B is equal to ∠A = ∠B in numerical terms. This is wrong, as a congruence (≅) means the side or angle of the geometric figures are the same and equality (=) is applicable for numerical values.
Mistake 4
Not considering the less than or equal in transitive property in inequality
Not considering the less than or equal in transitive property in inequality
Mistake 5
Thinking that transitive property does not apply to parallel lines
Students tend to believe that the transitive property does not apply to parallel lines, which is wrong. As the transitive property is applicable in geometry, if line p is parallel to line q, and line q is parallel to line r, then line p is parallel to line r.
Real-world applications of Transitive Property
The transitive property is a fundamental concept in mathematics, and it is used in different fields. Here are some real-world applications of the transitive property.
- Currency exchange rates in the international market are based on the transitive property. For example, if 1 USD = 80 INR and 1 Euro = 80 INR, then 1 USD = 1 Euro
- To solve algebraic equations in mathematics, we use transitive property, for instance, x = y and y = 5, then x = 5
- The transitive property is used to compare objects or geometric figures.
- In design, the equal measurements follow the transitive property. If one element is equal to the length of a second, and the second is equal to a third, then the first is equal to the third in length.
Solved Examples of Transitive Property
Problem 1
If 5 + 3 = 8 and 8 = 4 × 2, what can you conclude using the transitive property?
According to the transitive property, 5 + 3 = 4 × 2
Explanation
The transitive property states that if a = b and b = c, then a = c.
Therefore, 5 + 3 = 8 and 8 = 4 × 2, then 5 + 3 = 4 × 2
Problem 2
If ∠A = ∠B, and ∠B = ∠C, what can you conclude about ∠A and ∠C?
According to the transitive property ∠A = ∠C
Explanation
The transitive property states that if a = b and b = c, then a = c.
Here, ∠A = ∠B and ∠B = ∠C, then ∠A = ∠C
Problem 3
If triangle ABC is congruent to triangle DEF, and triangle DEF is congruent to triangle XYZ, what can you conclude about triangle ABC and triangle XYZ?
As per the transitive property, the triangle ABC is congruent to the triangle XYZ.
Explanation
The transitive property is also applicable to the congruence,
So, here the triangle ABC ≅ triangle DEF, and the triangle DEF ≅ triangle XYZ. Therefore, the triangle ABC ≅ to the triangle XYZ
Problem 4
If car A is faster than car B, and car B is faster than car C, what can you conclude about the speed of car A compared to car C?
Car A is faster than Car C
Explanation
According to the transitive property of inequality, if car A is faster than car B, and car B is faster than car C then car A is faster than Car C.
Problem 5
If x is less than or equal to y and y is less than or equal to z, what can you conclude about the relationship between x and z?
The relationship between x and z is x ≤ z
Explanation
The transitive property states that if a ≤ b, and b ≤ c, then a ≤ c
Here, if x ≤ y and y ≤ z, then x ≤ z
FAQs on Transitive Property
1.What is transitive property?
2.Is transitive property applicable to inequalities?
3.If x = y and y = z, then what is the relation between x and z?
4.What is the transitive property for parallel lines?
5.What are the applications of transitive property?
6.How can children in United States use numbers in everyday life to understand Transitive Property ?
7.What are some fun ways kids in United States can practice Transitive Property with numbers?
8.What role do numbers and Transitive Property play in helping children in United States develop problem-solving skills?
9.How can families in United States create number-rich environments to improve Transitive Property skills?
Hiralee Lalitkumar Makwana
About the Author
Hiralee Lalitkumar Makwana has almost two years of teaching experience. She is a number ninja as she loves numbers. Her interest in numbers can be seen in the way she cracks math puzzles and hidden patterns.
Fun Fact
: She loves to read number jokes and games.
