## Key Concepts

- Use algebraic properties of equality to form logical arguments.
- Use the properties in the real world.

### Algebraic Properties of Equality

When you solve an equation, you use the properties of real numbers. Segment lengths and angle measures are real numbers, so you can also use these properties to write logical arguments about geometric figures.

Let *a*, *b,* and *c* be real numbers.

**Addition Property: **If a = b, then a + c = b + c.

**Subtraction Property: **If a = b, then a – c = b – c.

**Multiplication Property: **If a = b, then ac = bc.

**Division Property:** If a = b and c ≠ 0, then a/c = b/c

**Substitution Property: **If a = b, then *a* can be substituted for *b* in any equation or expression.

**Distributive Property: **a(b + c) = ab + ac, where *a*, *b*, and *c* are real numbers.

### Properties of Equality in Geometry

The following properties of equality are true for all real numbers. Segment lengths and angle measures are real numbers, so these properties of equality are true for segment lengths and angle measures.

#### Symmetric property of equality

**Real numbers: **for any real numbers 𝑎 and 𝑏, if 𝑎 = 𝑏, then 𝑏 = 𝑎.

**Segment length: **for any segments

AB− and CD−, if 𝐴𝐵 = 𝐶𝐷, then 𝐶𝐷 = 𝐴𝐵.

**Angle measure: **for any angles ∠𝐴 and ∠𝐵, if 𝑚∠𝐴 = 𝑚∠𝐵, then 𝑚∠𝐵 = 𝑚∠𝐴.

#### Transitive property of equality

**Real numbers: **For any real numbers a, b and c, if a = b and b = c, then a = c.

**Segment length: **For any segments

AB, CD and EF, if AB = CD and CD = EF, then AB = EF.

**Angle measure: **For any angles ∠A, ∠B, and ∠C, if m∠A = m∠B and m∠B = m∠C, then

m∠A = m∠C.

#### Reflexive property of equality

**Real numbers:** for any real number a, a = a.

**Segment length: **for any segment

AB-, AB = AB.

**Angle measure: **for any angle ∠A, m∠a = m∠a.

### Let’s solve some examples!

#### Write reasons for each step

**Example 1: **Solve 2x + 5 = 20 – 3x. Write a reason for each step.

**Solution: **

#### Use the distributive property

**Example 2: **Solve –4(11x + 2) = 80. Write a reason for each step.

**Solution:**

#### Use properties in real-world

**Example 3: **

**HEART RATE:** When you exercise, your target heart rate should be between 50% to 70% of your maximum heart rate. Your target heart rate ** r** at 70% can be determined by the formula

*r = 0.70(220 – a),**where*

**represents your age in years. Solve the formula for**

*a*

**.**

*a***Solution:**

#### Use properties of equality

**Example 4:**

**Logo**: You are designing a logo to sell daffodils. Use the information given. Determine whether 𝑚∠𝐸𝐵𝐴 = 𝑚∠𝐷𝐵𝐶.

**Solution:**

#### Use properties of equality

**Example 5: **In the diagram, AB = CD. Show that AC = BD.** **

**Solution:**

### Questions to Solve

**Question 1**:

Name the property of equality the statement illustrates: If XY = AB and AB = GH, then XY = GH.

a. Substitution b. Reflexive c. Symmetric d. Transitive

**Solution:**

**d. Transitive **

**Question 2: **

Solve the equation. Write the reason for each step.

**4x + 9 = 16 – 3x **

**Solution:**

**Question 3:**

Show that the perimeter of triangle ABC is equal to the perimeter of triangle ADC.

**Solution:**

Given: AB = AD, CB = DC, AC = AC

Perimeter of triangle ABC = AB + AC + CB

Replacing AB as AD and CB as DC in the above equation,

Perimeter of triangle ABC = AD + AC + DC ** **

Since AD + AC + DC = perimeter of triangle ADC, perimeter of triangle ABC = perimeter of triangle ADC.

**Question 4**:

**Properties of equality **Copy and complete the table to show 𝑚∠2 = 𝑚∠3.

**Solution:**

**Key Concepts Covered **

### 1. Algebraic Properties of Equality

Let *a*, *b,* and *c* be real numbers.

**Addition Property: **If a = b, then a + c = b + c.

**Subtraction Property: **If a = b, then a – c = b – c.

**Multiplication Property: **If a = b, then ac = bc.

**Division Property:** If a = b and c ≠ 0, then a/c = b/c

**Substitution Property: **If a = b, then *a* can be substituted for *b* in any equation or expression.

**Distributive Property: **a(b + c) = ab + ac, where *a*,* b*, and *c* are real numbers.

### 2. Properties of Equality in Geometry** **

**Reflexive property of equality**

**Real numbers:** for any real number a, a = a.

**Segment length: **for any segment

AB- , AB = AB.

**Angle measure: **for any angle ∠A, m∠a = m∠a.

**Symmetric property of equality**

**Real numbers: **for any real numbers 𝑎 and 𝑏, if 𝑎 = 𝑏, then 𝑏 = 𝑎.

**Segment length: **for any segments

AB and CD, if 𝐴𝐵 = 𝐶𝐷, then 𝐶𝐷 = 𝐴𝐵.

**Angle measure: **for any angles ∠𝐴 and ∠𝐵, if 𝑚∠𝐴 = 𝑚∠𝐵, then 𝑚∠𝐵 = 𝑚∠𝐴.

**Transitive property of equality**

**Real numbers: **For any real numbers a,b and c, if a = b and b = c, then a = c.

**Segment length: **For any segments

AB , CD and EF, if AB = CD and CD = EF, then AB = EF.

**Angle measure: **For any angles ∠A,∠B, and ∠C, if m∠A = m∠B and m∠B = m∠C, then

m∠A = m∠C.

## Exercise

Solve the following equations for *x.*

- 5
*x*– 10 = –40 - 4
*x*+ 9 = 16 – 3*x* - 5(3x – 20) = –10
- 3(2x + 11) = 9
- 2x – 15 – x = 21 + 10x

Solve the following equations for y.

- 12 – 3y = 30x
- 3x + 9y = –7
- (1/2)x – (3/4)y = –2
- 3(w + 4) = 3w + 12 is an example of which property of equality?
- What is the division property of equality?

#### Related topics

#### Addition and Multiplication Using Counters & Bar-Diagrams

Introduction: We can find the solution to the word problem by solving it. Here, in this topic, we can use 3 methods to find the solution. 1. Add using counters 2. Use factors to get the product 3. Write equations to find the unknown. Addition Equation: 8+8+8 =? Multiplication equation: 3×8=? Example 1: Andrew has […]

Read More >>#### Dilation: Definitions, Characteristics, and Similarities

Understanding Dilation A dilation is a transformation that produces an image that is of the same shape and different sizes. Dilation that creates a larger image is called enlargement. Describing Dilation Dilation of Scale Factor 2 The following figure undergoes a dilation with a scale factor of 2 giving an image A’ (2, 4), B’ […]

Read More >>#### How to Write and Interpret Numerical Expressions?

Write numerical expressions What is the Meaning of Numerical Expression? A numerical expression is a combination of numbers and integers using basic operations such as addition, subtraction, multiplication, or division. The word PEMDAS stands for: P → Parentheses E → Exponents M → Multiplication D → Division A → Addition S → Subtraction Some examples […]

Read More >>#### System of Linear Inequalities and Equations

Introduction: Systems of Linear Inequalities: A system of linear inequalities is a set of two or more linear inequalities in the same variables. The following example illustrates this, y < x + 2…………..Inequality 1 y ≥ 2x − 1…………Inequality 2 Solution of a System of Linear Inequalities: A solution of a system of linear inequalities […]

Read More >>
Comments: