Question 489628
presumably these are all part of the same line.


you have the line of af equal to ab + bc + cd + de + ef


the line would look like this:


<pre>


         ---------------------------------------------------
         a         b         c         d         e         f

</pre>


you are given that the line segment ac is equal to df.
you are also given that the line segment bc is equal to ef


that would look like this:


<pre>

         ---------------------------------------------------------
         a         b         c               d         e         f



         ---------------------               ---------------------
         a                   c     equals    d                   f


                   -----------                         -----------
                   b         c         equals          e         f


</pre>


you are given that ac = df
you know that ac = ab + bc
you know that df = de + ef


this means that ab + bc = de + ef by substitution of equals.


you are given that bc = ef


this means that you can substitute bc for ef or you can substitute ef for bc in any equation.


the equation of interest is:


ab + bc = de + ef


substitute bc for ef to get:


ab + bc = de + bc


subtract bc from both sides of this equation to get:


ab = de


you can do this because of the basic algebraic property that says:


if a + c = b + c then a = b


this follows from the fact that if you subtract the same quantity from both sides of an equality, then the equality is preserved.


you are left with ab = de which is what you wanted to prove so you are done.