# Transitive Property

Peter, James, and David went for a walk in the park. James kept a track of the distance he covered. David and James were walking together and hence covered the same distance. Peter said that he and David had also walked the same distance. Can we thus conclude that the distance Peter covered is equal to the distance James recorded?

You will get a better understanding of the above relationship in this section.

The chapter will explore the transitive property meaning, transitive property of equality, transitive property of angles, and transitive property of inequality.

Check out the interactive simulations on the concept of transitive property and try your hand at solving a few interesting practice questions at the end of the page.

**Lesson Plan**

**What is Transitive Property?**

Transitive means to 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.**

If \(a\), \(b\), and \(c\) are three numbers such that \(a\) is equal to \(b\) and \(b\) is equal to \(c\), then \(a\) and \(c\) are equal to each other.

Let's assume Mary ate 2 hotdogs and Jake ate as many as Mary.

This indicates that they ate the same number of hotdogs.

Let's have a better understanding of the transitive property meaning and see its application.

**What is the General Formula of Transitive Property?**

The formula for the transitive property of equality is

If \(a = b \,\) and \(\,b = c\,\), then \(\,a = c\) |

Here \(a\), \(b\), and \(c\) are three quantities of the same kind.

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 transitive property, 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**

If we have angles \(m\) and \(n\) such that \(m=n\,\) and \(n=p\,\) and \(\,m= 40^{\circ}\), then by the transitive property of angles, we get \(p=40^{\circ}\)

**Transitive Property of Inequality for Real Numbers**

If we have three real numbers \(x\), \(y\), and \(z\) such that, \( x \leqslant y \,\) and\(\, y\leqslant z\,\), then\(\, x\leqslant z\).

Similarly, if we have a number \( p \leqslant 5 \,\) and\(\, 5\leqslant q\,\), then\(\, p\leqslant q\).

Use the transitive property of equality and inequality in the simulation below.

- 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.

**Solved Examples**

Example 1 |

The weight of a novel is same as the weight of a story book.

The story book 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, story book, and textbook be \(\,w_n, \,w_s, \,w_t.\)

The weight of the novel is the same as the weight of the story book. This indicates:

\(\,w_n=w_s\,\,....\) (Equation 1)

The story book weighs half the weight of the textbook. This indicates:

\(\,\,w_s=\dfrac{1}{2}\times\,w_t.\,\,....\) (Equation 2)

Combining Equation 1 and Equation 2, we get,

\begin{align} w_n&=\dfrac{1}{2}\times\,w_t...\text{(By Transitive Property)}\\&=\dfrac{1}{2}\times1.6\\&=0.8\\ \end{align}

\(\therefore\) Weight of the novel is 0.8 lb |

Example 2 |

Lucy is given the information that line p \(\left | \right |\)line q and line q \(\left | \right |\)line \(r\)

She wants to know the relation between angles \(a\) and \(b\)

Can you help her find the answer?

**Solution**

Since

\(line\, p \left | \right |line \,q\)

And

\( line \,q \left | \right |line \,r\)

By transitivity,

\(line\, p \,\left | \right |line\, r\)

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.

\(\therefore a = b\) |

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)

And

\( z=2y\,....\) (Equation 2)

Hence, by transitive property we get,

\begin{align} x+y&=2y\\x&= 2y-y\\\therefore x&=y \end{align}

\(\therefore x\,=\, y\) |

Can you find who is the tallest and who is the shortest?

Tim is taller than Tom. Tom is taller than Sam. Sam is taller than John. Jim is taller than Sam, but shorter than Tom. Mike is standing between Jack and Ted, who is shorter than Tim, but taller than Mike. Milo is standing between Tim and Leo, who is standing next to Ted. There are seven men standing between Milo and John.

**Interactive Questions**

**Here are a few activities for you to practice. **

**Select/Type your answer and click the "Check Answer" button to see the result.**

**Let's Summarize**

We hope you enjoyed learning about the **Transitive Property** with the simulations and practice questions. Now you will be able to easily solve problems on transitive property meaning, transitive property of equality, transitive property of angles, and transitive property of inequality.

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**Frequently Asked Questions (FAQs)**

## 1. 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.

## 2. When can you use transitive property?

We can use transitive property when we have three or more quantities of the same kind related by some rule.

## 3. What is the difference between transitive property and substitution property?

According to substitution property, when two things are equal, then one of them can replace the other in an expression.

If \(a = b\) and \(x= a-3,\) then \(x = b-3\)

According to transitive property, when two quantities are equal to the third quantity, then they are equal to each other.

If \(a = b\) and \(a = c,\) then \(b = c\)

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