Chapter 4.3, Problem 11WE

To determine

To find:Statement that are true.

  $(i)\overline{XW}\perp \overline{UV}(ii)\overline{UV}\left|\right|\overline{YZ}$

  $(iii)\overline{VX}\perp \overline{UX}$

Answer to Problem 11WE

True statement(s) are,

  $(i)\overline{XW}\perp \overline{UV}(ii)\overline{UV}\left|\right|\overline{YZ}$

Explanation of Solution

Giveninformation:In figure shown, $\overline{WX}\perp \overline{YZ}$ , $\angle 1\cong \angle 2$ , $\overline{UX}\cong \overline{VX}$ ,

  

Concept/Formula used:Two lines are parallel if their alternate interior angles are equal to each other.

Calculation: In given figure, as $\overline{WX}\perp \overline{YZ}$ , so,

  $\begin{array}{l}\angle WXY=\angle WXZ={90}^0\\ \Rightarrow \angle 1+\angle 3=\angle 2+\angle 4(\text{Given})\\ \Rightarrow \angle 1+\angle 3=\angle 1+\angle 4(\because \angle 1=\angle 2)\\ \Rightarrow \angle 3=\angle 4(\text{Subtraction}\text{property})\end{array}$

Againin $\Delta UXV$ ,

  $\begin{array}{l}\angle U+\angle V+\angle UXV={180}^0\\ \Rightarrow 2\angle U+\angle 3+\angle 4={180}^0(\because UX\cong VX,\Rightarrow \angle U\cong \angle V)\\ \Rightarrow 2\angle U+2\angle 3={180}^0\\ \Rightarrow \angle U+\angle 3={90}^0\end{array}$

Also in $\Delta UXW$ ,

  $\begin{array}{l}\angle U+\angle 3+\angle UWX={180}^0\\ \Rightarrow {90}^0+\angle UWX={180}^0\\ \Rightarrow \angle UWX={180}^0-90={90}^0\\ \Rightarrow \overline{WX}\perp \overline{UV}(\text{Definition of perpendicular lines)}\end{array}$

Now, as $\overline{WX}$ is perpendicular to both $\overline{UV}$ and $\overline{YZ}$ , $\overline{YZ}\left|\right|\overline{UV}$ .

Conclusion: Hence, the statements that are true are $\overline{XW}\perp \overline{UV}$ and $\overline{YZ}\left|\right|\overline{UV}$ .