Algebraic Properties

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

• 5x  – 10 = –40
• 4x  + 9 = 16 – 3x
• 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?