Chapter 7.4, Problem 31PPS

To determine

To write: The paragraph proof of Theorem $7.6$ .

Explanation of Solution

Given information:

The given Theorem is, “If a line intersects two sides of a triangle and separates the sides into proportional corresponding segments, then the line is parallel to the third side of the triangle”.

The given Theorem is, “If a line intersects two sides of a triangle and separates the sides into proportional corresponding segments, then the line is parallel to the third side of the triangle”.

Consider the following figure,

  

From the above statement, $\frac{BA}{CB}=\frac{DE}{CD}$ .

  $\begin{array}{l}\frac{BA}{CB}=\frac{DE}{CD}\\ 1+\frac{BA}{CB}=1+\frac{DE}{CD}\\ \frac{CB+BA}{CB}=\frac{CD+DE}{CD}\\ \frac{CA}{CB}=\frac{CE}{CD}\end{array}$

From SAS similarity postulate, $\Delta ACE\sim \Delta BCD$ and $\angle 1=\angle 4$ , so $\overline{BD}\left|\right|\overline{AE}$

Hence, if a line intersects two sides of a triangle and separates the sides into proportional corresponding segments, then the line is parallel to the third side of the triangle.