Last updated on July 5th, 2025
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.
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.
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:
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.
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.
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.
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.
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.
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
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
If ∠A = ∠B, and ∠B = ∠C, what can you conclude about ∠A and ∠C?
According to the transitive property ∠A = ∠C
The transitive property states that if a = b and b = c, then a = c.
Here, ∠A = ∠B and ∠B = ∠C, then ∠A = ∠C
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.
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
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
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.
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
The transitive property states that if a ≤ b, and b ≤ c, then a ≤ c
Here, if x ≤ y and y ≤ z, then x ≤ z
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.
: She loves to read number jokes and games.