Question

given: ( ac = bd )
prove: ( ab = cd )

statement	reason
2 ( bd = bc + cd )	addition and subtraction properties
3 ( ac = bd )	given
4 ( ab + bc = bc + cd )	reason?
5 ( ab = cd )	reason?

Explanation:

Step1: Substitute using given equalities

Since \(AC = AB + BC\) (Statement 1), \(BD=BC + CD\) (Statement 2), and \(AC = BD\) (Statement 3), we substitute \(AC\) and \(BD\) in the equation. So, \(AB + BC=BC + CD\) (by substitution property of equality).

Step2: Subtract \(BC\) from both sides

Using the subtraction property of equality (\(a + c=b + c\Rightarrow a = b\)), if \(AB + BC=BC + CD\), then \(AB+BC - BC=BC + CD - BC\). Simplifying gives \(AB = CD\).

Answer:

1. Substitution Property of Equality; 5. Subtraction Property of Equality