a Let's illustrate the problem and label the different locations accordingly.
We can now use these labels to write the given information about the lengths.
Given:& RS= CF,
& SM= MC= FD
We are asked to prove that the distance from the restaurant to the movie theater RM is the same as the distance from the cafe to the dry cleaners CD.
Prove: RM=CD
b Let's begin with reviewing the idea of a two-column proof. It lists the statements on the left column and their corresponding justifications on the right column. Each statement must follow logically from the previous steps. Let's start with what we found on Part A!
Statement1)& RS= CF, & SM= MC= FD Reason1)& Given
Since points R, S, and M are collinear and S is between R and M, we can use the Segment Addition Postulate. We can do the same for points C, F, and D.
This will be our second statement.
Statement& 2) RM= RS+SM, & CD=CF+FD Reason& 2) Segment Addition Postulate
We are given that RS= CF and SM= FD. By the Substitution Property of Equality, we can substitute RS and SM for CF and FD, respectively, in the diagram.
Therefore, CD is the sum of RS and SM. This is our third statement.
Statement& 3) CD=RS+SM Reason& 3) Substitution Property & of Equality
Now we have that both RM and CD are equal to RS+ SM. This means that, by the Transitive Property of Equality, RM is equal to CD.
Statement& 4) RM=CD Reason& 4) Transitive Property & of Equality
Let's write this as a two-column proof!
