Chapter 6.3, Problem 6PSB

To determine

Prove that $\overline{\mathrm{A}\mathrm{B}}\cong \overline{\mathrm{C}\mathrm{D}}$ .

Answer to Problem 6PSB

  $\overline{\mathrm{A}\mathrm{B}}\cong \overline{\mathrm{C}\mathrm{D}}$ Proved.

Explanation of Solution

Given information:

Given: $\mathrm{m}\parallel \mathrm{n}$ , $\overleftrightarrow{\mathrm{A}\mathrm{B}}\parallel \overleftrightarrow{\mathrm{C}\mathrm{D}}$

  

Calculation

Consider the given figure,

  

It is given that,

  $\begin{array}{l}\mathrm{m}\left|\right|\mathrm{n}\\ \overline{\mathrm{A}\mathrm{C}}\mathrm{l}\mathrm{i}\mathrm{e}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{m}\overline{\mathrm{B}\mathrm{D}}\mathrm{l}\mathrm{i}\mathrm{e}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{n}\overline{\mathrm{A}\mathrm{B}}\left|\right|\overline{\mathrm{C}\mathrm{D}}\end{array}$

We know that parallel lines determine a plane so,

  $\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{D}\mathrm{i}\mathrm{s}\mathrm{a}\mathrm{p}\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{e}$

If a plane intersects two lines the lines are parallel.

  $\overline{\mathrm{A}\mathrm{C}}\left|\right|\overline{\mathrm{B}\mathrm{D}}$

  $\mathrm{A}\mathrm{B}\mathrm{C}\mathrm{D}$ is a parallelogram, if a quadrilateral whose two pairs of opposite sides are parallel is a parallelogram.

The opposite sides of parallelogram are congruent.

Hence, proved $\overline{\mathrm{A}\mathrm{B}}\cong \overline{\mathrm{C}\mathrm{D}}$ .