Chapter 7.6, Problem 9CE

To determine

To Prove: The line $a,b,$ and $c$ are parallel with each other with the help of given image.

Explanation of Solution

Given information:

  

Corollary:

If two transversals are intersected by three lines, they divide the transversals proportionally.

If $\overrightarrow{RX}\parallel \overrightarrow{SY}\parallel \overrightarrow{TZ}$

Then, $\frac{RS}{ST}=\frac{XY}{YZ}$

And the converse of the corollary of the triangle proportionality theorem is: if three lines divide two transversals proportionally, then the lines are parallel. Yes, the converse is true.

To prove that, consider three parallel line, $a,b$ and $c$ which are intersected by one line at points $A,B,$ and $C$ respectively and by another at $D,E,$ and $F$ respectively and by another at $G,F,$ and E respectively.

Now, if three lines divide three transversals proportionally then,

  $\frac{AB}{BC}=\frac{AD}{DE}=\frac{GF}{FE}$

  $\frac{16}{24}=\frac{18}{27}=\frac{20}{30}$

  $\frac{2}{3}=\frac{2}{3}=\frac{2}{3}$

As the ratio of the all corresponding segments are equal.

Hence, the lines $a,b,$ and $c$ are parallel with each other.