identify the properties

1.) if a=b and c=d then ac=bd
2.) c(d+e) = cd+ce
3.) if c=d and d=e ten c=e
4.) if x=y and p=q then x+p=y+q
5.) if t=r then r=t

To identify the properties used in these statements, we can break down the equations and recognize the mathematical concepts involved. Here is a breakdown for each property:

1.) Property: Multiplication Property of Equality
Explanation: This property states that when two equal quantities (a and b) are multiplied by the same quantity (c and d), their product (ac and bd) will also be equal. This property allows us to multiply both sides of an equation by the same number without affecting the equality.

2.) Property: Distributive Property
Explanation: The distributive property states that when a number (c) is multiplied by the sum of two other numbers (d + e), it is equivalent to multiplying the number by each of the two numbers and adding the products together (cd + ce). This property is fundamental to simplify expressions involving multiplication and addition.

3.) Property: Transitive Property of Equality
Explanation: The transitive property states that if two quantities (c and d) are equal, and the second quantity (d) is also equal to a third quantity (e), then the first quantity (c) must be equal to the third quantity (e). This property allows us to establish a chain of equality when multiple equalities are involved.

4.) Property: Addition Property of Equality
Explanation: The addition property of equality states that if two quantities (x and y) are equal, and another two quantities (p and q) are also equal, then the sum of the first two quantities (x + p) is equal to the sum of the second two quantities (y + q). This property allows us to add or subtract the same number to both sides of an equation without changing the equality.

5.) Property: Symmetric Property of Equality
Explanation: The symmetric property of equality states that if two quantities (t and r) are equal, then the order of the quantities does not matter. In other words, t = r implies r = t. This property allows us to reverse the order of equalities without affecting their validity.

By understanding these properties, we can apply them to solve equations, simplify expressions, and establish relationships between quantities.