We want to check whether Carolina's proof is in a logical sequence to prove that a=d. This means that all statements have to be either given or be implied by the previous steps. Let's start with the first one.
Statement
Reason
1.
AD∥ EH
1.
Given
The first statement is given in the exercise — it can be recalled at any moment in the proof. Let's move to the second statement.
Statement
Reason
1.
AD∥ EH
1.
Given
2.
a=b
2.
If lines are parallel alternate interior angles are equal
The second statement also follows a logical sequence, as we can deduce it from the first one. Now, let's move to the third one.
Statement
Reason
1.
AD∥ EH
1.
Given
2.
a=b
2.
If lines are parallel alternate interior angles are equal
3.
a=c
3.
Substitution
Now we see that in the third row a and c are equated. We can do this because b=c and then use the Transitive Property of Equality.
a= b, and b= c ⇒ a= c
However, before we can do this we have to state that b=c by the Corresponding Angles theorem. In Carolina's proof this part is the fifth statement, which breaks the chain of logic. With this information we can rewrite the proof so that it comes in a logical order.
Statement
Reason
1.
AD∥ EH
1.
Given
2.
a=b
2.
If lines are parallel alternate interior angles are equal
3.
BF∥ CG
3.
Given
4.
b=c
4.
If lines are parallel, corresponding angles are equal
