Chapter 6.3, Problem 7PSB

To determine

Prove that $\overline{\mathrm{R}\mathrm{V}}\cong \overline{\mathrm{S}\mathrm{T}}$ .

Answer to Problem 7PSB

  $\overline{\mathrm{R}\mathrm{V}}\cong \overline{\mathrm{S}\mathrm{T}}$ Proved.

Explanation of Solution

Given information:

Given: $\mathrm{e}\parallel \mathrm{f}$ , $\overline{\mathrm{R}\mathrm{T}}\cap \overline{\mathrm{V}\mathrm{S}}=\mathrm{P}$ , $\overline{\mathrm{R}\mathrm{S}}\cong \overline{\mathrm{V}\mathrm{T}}$

  

Calculation

Consider the given figure,

  

It is given that,

  $\overline{\mathrm{R}\mathrm{T}}\cap \overline{\mathrm{V}\mathrm{S}}=\mathrm{P}$ , $\mathrm{c}\left|\right|\mathrm{f}$ , $\overline{\mathrm{R}\mathrm{S}}\cong \overline{\mathrm{V}\mathrm{T}}$

We know that parallel lines determine a plane so,

  $\mathrm{R}\mathrm{V}\mathrm{T}\mathrm{S}$ is a plane

If a plane intersects two lines the lines are parallel.

  $\overline{\mathrm{R}\mathrm{S}}\left|\right|\overline{\mathrm{V}\mathrm{T}}$

  $\mathrm{R}\mathrm{V}\mathrm{T}\mathrm{S}$ 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 $\overline{\mathrm{R}\mathrm{V}}\cong \overline{\mathrm{S}\mathrm{T}}$ .