Transitive Property of Equality
Transitive property of equality states that if two numbers are equal to each other and the second number is equal to the third number, then the first number is also equal to the third number. In other words, a = b, b = c, then a = c. The transitive property of equality is one among the many properties of equality in math. This chapter will explore the transitive property meaning, transitive property of equality, transitive property of angles, and transitive property of inequality.
What is Transitive Property of Equality?
The word transitive means transfer. If a, b, and c are three quantities, and if a is related to b by some rule and b is related to c by the same rule, then a and c are related to each other by the same rule, this property is called transitive property of equality. Let us consider the quantities a = b and b = c. According to the symmetric property of equality, writing a = b is the same as b = a. Hence, we can say that b = a and b = c. However, the quantity b cannot be equal to two different quantities and a must be equal to c. Therefore, we can say that a = c. Let us look at an example, assume Mary ate 2 hotdogs and Jake ate as many as Mary. This indicates that they ate the same number of hotdogs, according to the transitive property of equality.
General Formula of Transitive Property
The formula for the transitive property of equality is: If a = b, b = c, then a = c. Here a, b, and c are three quantities of the same kind. This property holds good for real numbers. For example, if a is the measure of an angle, then b or c can't be the length of the segment.
Example: If x = m and m = 7, then we can say x = 7
The value 7 is transferred to x because x and m are equal.
How to Use Transitive Property?
To use the transitive property of equality, we need three or more quantities for relating. This property can be extended to the transitive property of angles and transitive property of inequality as well.
Transitive Property of Angles
According to the transitive property of congruence, if any two angles, lines, or shapes are congruent to a third angle, line, or shape respectively, then the first two angles, lines, or shapes are also congruent to the third angle, line, or shape. For example, if we have angles m and n such that m = n and n = p and m = 40°, then by the transitive property of angles, we get p = 40°.
Transitive Property of Inequality of Real Numbers
Transitive property applies to inequality as well and it states that, if we have three real numbers x, y, and z such that, x ⩽ y and y ⩽ z, then x ⩽ z. For example, if we have a number p ⩽ 5 and 5 ⩽ q, then p ⩽ q. Let us look at another example, consider three people whose weights are unknown. It is known that Sam weighs lesser than Rob and that Rob weighs lesser than Charlie. According to the property, we can say that Sam also weighs lesser than Charlie.
Proof of Transitive Property of Equality
This property cannot be proved as it is an axiom. However, one of the wellknown examples to prove the transitive property of equality is by constructing an equilateral triangle using a ruler and compass. Hence, the aim is to show that the object constructed is indeed an equilateral triangle with the help of the property.
Construct a line segment AB with any measurement. Draw two circles where one circle crosses point A and the other circle crosses point B. Where for one circle A is the center and AB is the radius and the other circle has B as the center and BA as the radius. Mark the intersection of the two circles as C. Complete the triangle by connecting A to C and B to C creating ABC. As we know that AB is the radius of one circle with A as the center, we can also consider AC as the radius of this circle making all the radii equal hence AB = AC (see yellow circle). AB can also be considered as the radius for the other circle with center B because according to the reflexive property of addition, BA = AB. And for the same circle, BC can also be considered as a radius hence AB = BC (see blue circle). Since AB = AC and AB = BC then we can say AC = BC according to the transitive property of equality. Therefore, all three lines are equal to each other making ABC an equilateral triangle.
Important Notes
 The transitive property of equality : If a = b and b = c, then a = c
 This property can be applied to numbers, algebraic expressions, and various geometrical concepts like congruent angles, triangles, circles, etc.
Related Topics
Listed below are a few topics related to transitive property of equality, take a look.
Transitive Property of Equality Examples

Example 1: The weight of a novel is the same as the weight of a storybook. The storybook weighs half the weight of a textbook. If the weight of the textbook is 1.6 lb, what is the weight of the novel?
Solution:
Let the weight of the novel, storybook, and textbook be (w)_{n}, (w)_{s}, (w)_{t}
The weight of the novel is the same as the weight of the storybook. This indicates:
(w)_{n} = (w)_{s} (Equation 1)
The storybook weighs half the weight of the textbook. This indicates:
(w)_{s} = 1/2 × (w)_{t} (Equation 2)
Combining Equation 1 and Equation 2, we get,
(w)_{n} = 1/2 × (w)_{t} (by transitive property)
= 1/2 × 1.6
= 0.8
Therefore, the weight of the novel is 0.8 lb.

Example 2: Lucy is given the information that line p II line q and line q II line r. She wants to know the relation between angles a and b. Can you help her find the answer?
Solution: Since, line p II line q and line q II line r, therefore, by transitive property of equality line p II line r.
We know that when two lines are parallel, the corresponding angles are equal. Here, a and b are corresponding angles. Hence, they have equal measure.
Therefore, angle a = angle b.

Example 3: Susan gives two hints to Mike and challenges him to find the relation between x and z. Hints : x+ y = z, z = 2y. Let's find out how Mike can complete this task.
Solution: Given,
x + y = z (Equation 1)
z = 2y (Equation 2)
By transitive property of equality we get,
x + y = 2y
x = 2y  y
x = y
Therefore, x = y.
Practice Questions on Transitive Property of Equality
FAQs on Transitive Property of Equality
What is Transitive Property Proof?
The transitive property is an axiom. That means, it is a universally accepted truth. Hence, we don't need to prove this property.
When Can You Use Transitive Property?
We can use the transitive property when we have three or more quantities of the same kind related by some rule.
What is the Difference Between Transitive Property and Substitution Property?
According to the substitution property, when two things are equal, then one of them can replace the other in an expression i.e. a = b and x = a−3, then x = b−3
According to the transitive property, when two quantities are equal to the third quantity, then they are equal to each other i.e. a = b and a = c, then b = c
How Do You Remember Transitive Property of Equality?
Transitive property of equality can be seen among parallel lines along with numbers and equations.
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