Properties of Equality
Easily Explained w/ 9 Examples!

To prove equality and congruence, we must use sound logic, properties, and definitions. Otherwise known as properties of equality.

By knowing these logical rules, we will be able to manipulate, simplify, balance, and solve equations, as well as draw accurate conclusions supported by reasons.

The following properties allow us to simplify, balance, and solve equations.

Algebraic Properties Of Equality

1. Addition

Definition

If a = b, then a + c = b + c

Example

If x – 3 = 7, then x = 10 by adding 3 on both sides.

2. Subtraction

Definition

If a = b, then a – c = b – c

Example

If x + 2 = 11, then x = 9 by subtracting 2 on both sides.

3. Multiplication

Definition

If a = b, then ac = bc

Example

If x/4 = 5, then x = 20 by multiplying by 4 on both sides.

4. Division

Definition

If a = b and c is not equal to 0, then a / c = b / c

Example

If 8x = 16, then x = 2 by dividing by 8 on both sides.

5. Distributive

Definition

If a (b + c), then a (b + c) = ab + ac

Example

If 3(x + 2), then 3(x + 2) = 3x + 6 by multiplying both x and 2 by 3.

6. Substitution

Definition

If a = b, then either a or b may be substituted into any algebraic equation or expression for the other.

Example

If 5x – 2y = z and x = y, then 5x – 2x = 12 or 5y – 2y = 12 by the substitution property.

And these three properties define an equivalence relation, as accurately stated by Varsity Tutors.

In the video below, you’ll learn to use these properties of equality, along with our previously learned definitions and postulates, to draw conclusions.

We’ll start by showing you how to generate reasons for solving basic algebraic equations and then progress to supplying missing statements or reasons for equality and congruence involving geometric figures like segments and angles.

All in all, we are discovering and learning how to use our final tool in our tool belt, thus enabling us to tackle two-column proofs in the next lesson.

Properties of Equality – Lesson & Examples (Video)

38 min

• Introduction to properties of equality and congruence
• 00:00:36 – Algebraic properties of equality and congruence (Examples #1-2)
• Exclusive Content for Member’s Only

• 00:13:26 – What are the missing reasons in the proof (Examples #2-4)
• 00:19:52 – Fill in the blanks to complete the two-column proof (Examples #5-7)
• 00:29:57 – Fill in the missing statements or reasons for the following two-column proof (Examples #8-9)
• Practice Problems with Step-by-Step Solutions
• Chapter Tests with Video Solutions