Chapter 7.6, Problem 8CE

To determine

To prove: The lines are parallel with each other and show the converse is true.

Answer to Problem 8CE

The lines are parallel with each other does the converse is true.

Explanation of Solution

Given information:

The converse of the corollary of the triangle proportionality theorem is:

If three lines divide two transversals proportionally, then the lines are parallel.

Calculation:

Corollary: The transversals are proportionally dividing, if three lines intersect two transveral.

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.

Now, if three lines divide two transversals proportionally then,

  $\begin{array}{l}\frac{AB}{BC}=\frac{DE}{EF}\\ \frac{6}{8}=\frac{9}{12}\\ \frac{3}{4}=\frac{3}{4}\\ L\cdot H\cdot S=R\cdot H\cdot S\end{array}$

Hence, the converse is true.