Question

introduction to proofs: justifying statements
mplete the proof by choosing the correct reason.
given: ( a c = b d )
prove: ( a b = c d )

Explanation:

Step1: Segment Addition Postulate

The Segment Addition Postulate states that if \(B\) is between \(A\) and \(C\), then \(AC = AB+BC\).

Step2: Segment Addition Postulate

Similarly, if \(C\) is between \(B\) and \(D\), by the Segment Addition Postulate, \(BD = BC + CD\).

Step3: Given

The problem statement provides \(AC = BD\) as given information.

Step4: Substitution Property of Equality

Since \(AC = AB + BC\), \(BD=BC + CD\), and \(AC = BD\), we substitute \(AC\) with \(AB + BC\) and \(BD\) with \(BC + CD\) in the equation \(AC = BD\), getting \(AB + BC=BC + CD\).

Step5: Subtraction Property of Equality

Subtract \(BC\) from both sides of the equation \(AB + BC=BC + CD\). Using the Subtraction Property of Equality (\(a + c=b + c\Rightarrow a = b\)), we get \(AB=CD\).

Answer:

1. Segment Addition Postulate
2. Segment Addition Postulate
3. Given
4. Substitution Property of Equality
5. Subtraction Property of Equality