Equality, Distributive, and Congruence Properties
Standard
Prepares for G.CO.10 Prove theorems about triangles...
Goals for this section:
This section is largely review. Students should be able to recall basic properties of Algebra such as the Properties of Equality, the Distributive Property, and Properties of Congruence.
Properties of Equality
Addition Property
If a = b, then a + c = b + c.
As long as you add on both sides, the equation is still true!
Subtraction Property
If a = b, then a - c = b - c.
As long as you subtract on both sides, the equation is still true!
Multiplication Property
If a = b, then a * c = b * c.
As long as you multiply on both sides, the equation is still true!
Division Property
If a = b, then a / c = b / c.
As long as you divide on both sides, the equation is still true!
Reflexive Property
a = a
Something is reflexive when you say it equals itself.
Symmetric Property
If a = b, then b = a.
The order doesn't matter, they're still equal to each other.
Transitive Property
If a = b and b = c, then a = c.
By the power of logic, a must equal c!
Substitution Property
If a = b, then b can replace any instance of a in an expression.
This is what really makes the transitive property a thing.
The Distributive Property
Distributing is multiplying the left terms to all of the right terms.
a(b + c) = ab + ac
(a + b)(c + d) = ac + ad + bc + bd
Properties of Congruence
Reflexive Property
AB ≅ AB
∠A≅∠A
Symmetric Property
If AB ≅ CD, then CD ≅ AB
If ∠A≅∠B, then ∠B≅∠A
Transitive Property
If AB ≅ CD and CD ≅ EF, then AB ≅ EF
If ∠A ≅ ∠B and ∠B ≅ ∠C, then ∠A ≅ ∠C
