Chapter 2.5, Problem 22WE

To determine

To check: if two line segments are perpendicular to each other

Answer to Problem 22WE

Yes, it can be concluded $\overrightarrow{XY}\perp \overrightarrow{XZ}$ .

Explanation of Solution

Given information:

A diagram representing section of line segments with given interior angles such that $\angle 1$ and $\angle 3$ are congruent and complimentary to each other.

Formula used:

1) Two angles become congruent when they are equal to one another.

  $\angle A=\angle B$

2) Two angles are complimentary to each other when their sum is equal to 90⁰$\angle A+\angle B=90°$

3) Two line segments are perpendicular to each other when their included angle is 90⁰.

Calculation:

Consider the figure given below.

  

Based on the angles shown in the figure, it can be expressed as,

  $\angle 1+\angle 2+\angle 3+\angle 4=180°$ ...... (1)

Now, given that the angles $\angle 1$ and $\angle 3$ are congruent as well as complimentary, it can be expressed as,

  $\begin{array}{l}\angle 1=\angle 3\\ \angle 1+\angle 3=90°\end{array}$

This equation can be rewritten to obtain the value of $\angle 1$ and $\angle 3$ as,

  $\begin{array}{l}2\angle 1=2\angle 3=90°\\ \angle 1=\angle 3=45°\end{array}$ ......…. (2)

Also, the segment $\overrightarrow{XY}$ is perpendicular to the line through the X such that,

  $\angle 1+\angle 2=90°$

Using this relation and the fact that $\angle 1=45°$ , the angle 2 can be written as,

  $\begin{array}{c}\angle 2=90°-\angle 1\\ =90°-45°\\ =45°\end{array}$ ......…. (3)

Thus from (2) and (3),the relation between angles 2 and 3 is given by,

  $\angle 2+\angle 3=90°$

It can be observed from the figure that, the sum of the angles $\angle 2$ and $\angle 3$ is the included angle between the line segments $\overrightarrow{XY}$ and $\overrightarrow{XZ}$ which has been found to be 90⁰. Thus it can be concluded $\overrightarrow{XY}\perp \overrightarrow{XZ}$ .