Question

given: eg = hk, fg = hj. prove: ef = jk. 5. ef + fg = hj + jk, reason: substitution property. 6. ef · fg = hj · jk, reason: substitution property. 7. validate, reason?

Explanation:

Step1: Recall segment - addition postulate

By the segment - addition postulate, \(EG=EF + FG\) and \(HK = HJ+JK\).

Step2: Use given information

Given \(EG = HK\), \(FG=HJ\). Substitute \(EG\) with \(EF + FG\) and \(HK\) with \(HJ + JK\) in the equation \(EG = HK\), we get \(EF+FG=HJ + JK\).

Step3: Apply substitution property

Since \(FG = HJ\), we can substitute \(FG\) with \(HJ\) in the equation \(EF+FG=HJ + JK\). After substitution, we subtract \(FG\) (or \(HJ\)) from both sides of the equation \(EF+FG=HJ + JK\) and obtain \(EF=JK\).

Answer:

The proof is valid. The reason for step 7 is the Subtraction Property of Equality (after using substitution to get an equation where we can subtract the equal - length segments from both sides of the equation \(EF + FG=HJ + JK\) since \(FG = HJ\)).