Download a=b

Reason Using Properties from Algebra
2.5
Looking back at algebra, we used different properties of real numbers to solve an
equation. These properties include:
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, then
Substitution Property
If a=b, then a can be substituted for b in
any equation or expression
𝑎
𝑐
=
𝑏
𝑐
Example: Solve 2𝑥 + 5 = 20 − 3𝑥 and write a reason for each step.
2𝑥 + 5 = 20 − 3𝑥
5𝑥 + 5 = 20
5𝑥 = 15
𝑥=3
Given
Addition Property of Equality
Subtraction Property of Equality
Division Property of Equality
We can also use the Distributive Property: 𝑎(𝑏 + 𝑐) = 𝑎𝑏 + 𝑎𝑐 where a, b, and c are
real numbers.
Example: Solve −4(11𝑥 + 2) = 80 and write a reason for each step.
−4(11𝑥 + 2) = 80
(11𝑥 + 2) = −20
11𝑥 = −22
𝑥 = −2
Properties of Equality (true for all real numbers)
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
Example: (Use the properties of equality) You are designing a logo to sell daffodils.
Use the information given. Determine whether 𝑚∠EBA = m∠DBC
𝑚∠1 = 𝑚∠3
𝑚∠𝐸𝐵𝐴 = 𝑚∠2 + 𝑚∠3
𝑚∠𝐸𝐵𝐴 = 𝑚∠2 + 𝑚∠1
𝑚∠1 + 𝑚∠2 = 𝑚∠𝐷𝐵𝐶
𝑚∠𝐸𝐵𝐴 = 𝑚∠𝐷𝐵𝐶