To solve this problem and make sense of the provided statements and proof steps, let's walk through the reasoning and identify the appropriate property at each stage.
We are given the following information:
1. [tex]\(WX = YZ\)[/tex] (Given)
2. [tex]\(XY = XY\)[/tex] (Reflexive Property of Equality)
3. [tex]\(WX + XY = XY + YZ\)[/tex] (This uses the Addition Property of Equality)
4. [tex]\(X\)[/tex] is between [tex]\(W\)[/tex] and [tex]\(Y\)[/tex]. (Given)
5. [tex]\(WX + XY = WY\)[/tex] (Definition of between)
6. [tex]\(WY = XY + YZ\)[/tex] (We need a reason here)
7. [tex]\(Y\)[/tex] is between [tex]\(X\)[/tex] and [tex]\(Z\)[/tex]. (Given)
8. [tex]\(XY + YZ = XZ\)[/tex] (Definition of between)
9. [tex]\(WY = XZ\)[/tex] (Substitution Property of Equality)
Now, let’s focus on Statement 3, [tex]\(WX + XY = XY + YZ\)[/tex].
Explanation:
- This statement follows from the principle that if you add the same quantity to both sides of an equation, the equality is preserved. Here, [tex]\(XY\)[/tex] is added to both sides of the equality [tex]\(WX = YZ\)[/tex].
- By adding [tex]\(XY\)[/tex] to both sides of the initial equation we get:
[tex]\[
WX + XY = YZ + XY.
\][/tex]
The correct property that justifies this step is the Addition Property of Equality, which states that you can add the same quantity to both sides of an equation and still maintain equality.
Thus, the reason for Statement 3 is:
Addition Property of Equality
