Transitive Property

In mathematics, the transitive property is one of the properties of equality. The word “transitive” refers to 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.

For example, if we have the equal number of pens and pencils, and equal number of pencils and books, then according to the transitive property, the number of pens are the same as the number of books.

The transitive property states if a = b and if 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: The reflexive property states that any real number is equal to itself, i.e., a = a.
   
• Symmetric property of equality: The symmetric property states that if one quantity equals another, the order of their equality does not matter. That is, if a = b, then b = a.

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

That is, 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, then a ≤ c.

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 each congruent to a third shape, 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 various 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 if 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 also applicable in the 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.
   
• For the circle with center C, the radii 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.

Here are some practical tips and tricks to master the transitive property of equality in mathematics.
 

• Always check whether the related quantities share the same units before applying the property of transitivity.
   
• The transitive property helps us in algebraic manipulation and to prove geometric concepts such as congruence or similarity.
   
• Practice with real-world scenarios like comparing numbers, weights, distances to see the property of transitivity work.
   
• Try to combine the reflexive property and symmetric property to simplify proofs and calculations.
   
• Solve many problems using the transitivity property in both algebra and geometry to reinforce conceptual clarity.

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 approximately.
   
• 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. 
   
• In computer science, sorting algorithms rely on transitivity, and we can compare databases and objects using transitivity.