Question

step statement reason
1 ac = ab + bc addition and subtraction properties
2 de = db + be substitution property
3 ac = de given
4 ab + bc = db + be reflexive property
5 ab = be angle addition property
6 be + bc = db + be multiplication and division properties
7 bc = db addition and subtraction properties

Explanation:

Step1: Analyze each step's reason

• Step 1: \(AC = AB + BC\) is based on the Segment Addition Postulate (which is related to Addition and Subtraction Properties in a geometric sense).
• Step 2: \(DE=DB + BE\) is also based on the Segment Addition Postulate (not Substitution Property).
• Step 3: \(AC = DE\) is Given.
• Step 4: Since \(AC=AB + BC\), \(DE = DB + BE\) and \(AC = DE\), by Substitution Property \(AB + BC=DB + BE\) (not Reflexive Property).
• Step 5: If \(AB + BC=DB + BE\) and we want to get to \(BE+BC = DB + BE\) (step 6), we use Substitution Property (substituting \(AB\) with \(BE\) if \(AB = BE\)). But the reason for \(AB = BE\) is not given in the problem context. If we assume it's a given or derived correctly, then step 6: \(BE+BC = DB + BE\) is substitution. And step 7: Subtract \(BE\) from both sides of \(BE + BC=DB + BE\) (by Addition and Subtraction Properties) to get \(BC = DB\).

Answer:

The correct reasons (assuming \(AB = BE\) is a valid given or derived step):

1. Segment Addition Postulate (related to Addition and Subtraction Properties)
2. Segment Addition Postulate
3. Given
4. Substitution Property
5. (Reason depends on problem - if \(AB = BE\) is given, then Given; if derived, state the derivation reason)
6. Substitution Property
7. Addition and Subtraction Properties (Subtraction of \(BE\) from both sides)