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 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. 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: 

• 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 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. 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 is less than or equal to the third, 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 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.
   
• 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. 
   

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.