Chapter 3.5, Problem 39HP

(a)

To determine

Construct the line parallel to $\overline{BC}$ through point A in the triangle ABC.

Explanation of Solution

Calculation :

The Diagram:

  

ABC is a triangle and $\overline{BC}\parallel l$

(b)

To determine

Show that $\overline{BC}\parallel l$ using measurement.

Explanation of Solution

Calculation :

From part (a), the Diagram:

  

Draw the height of the triangle .It is equidistant from $\overline{BC}\text{ and }l$ . So, $\overline{BC}\parallel l$

(c)

To determine

Show that $\overline{BC}\parallel l$ in part (a) .

Explanation of Solution

Calculation :

From part (a), the Diagram is given .

Copy and paste the vertex B of triangle on the vertex A :

  

  $\overline{AB}$ is transversal for $\overline{BC}\text{ and }\overline{AE}$ .

Since triangle ABC was copied to construct triangle DAE ,

  $\begin{array}{l}\Delta ABC\cong \Delta DAE\\ \Rightarrow \angle ABC\cong \angle DAE\end{array}$

and they are the corresponding angles.

So, by the Converse of Corresponding Angle Theorem which states that when a transversal intersects two lines , the two lines are parallel if the corresponding angles are congruent, $\overline{BC}\parallel \overline{AE}$ .

So, $\overline{BC}\parallel l$

Hence proved.