Chapter 2.3, Problem 12WE

To determine

To sketch :  A geometrical figure which is same as the one given below.

  

Explanation of Solution

Given information :  a geometricalfigure as shown:

  

Graph :  Draw a line and mark points L and Nas vertex. Mark a point Msomewhere on line LN . Take a protractor and place its horizontal edge on LNand center on M . Measure $\angle$ PMN from the given figure. Start from zero on the right, read the scale on the protractor and mark the reading equal to $\angle$ PMN. Mark the point P and join $\overrightarrow{MP}$ and thus resulting figure is constructed as below,

  

Interpretation :  The resulting figure is same as the given figure.

To determine

To sketch :  The angle bisectors of $\angle$ PMN and $\angle$ LMP by the use of a protractor. The angles $\angle$ PMN and $\angle$ LMP are shown in the figure below,

  

Explanation of Solution

Given information :  Given are two angles $\angle$ PMN and $\angle$ LMP as shown below, whose bisectors are to be constructed by the use of a protractor.

  

Graph :  Take a protractor and place its horizontal edge on LNand center on M . Measure $\angle$ LMP from the given figure. Take half of the reading. Start from ${180}^0$ on the left, subtract $\frac{1}{2}\angle$ LMP from ${180}^0$ and mark O onthat value. $\overrightarrow{MO}$ is the angle bisector of $\angle$ LMPas shown below,

  

Again, Measure $\angle$ PMN from the given figure. Start from zero on the right, read the scale on the protractor, take half of the reading of $\angle$ PMN , and mark that point as Q . $\overrightarrow{MQ}$ is the angle bisector of $\angle$ PMN as shown below,

  

Thus the two angle bisectors are shown in the figure below.

Interpretation :  $\overrightarrow{MQ}$ and $\overrightarrow{MO}$ are the angle bisectors of angles $\angle$ PMNand $\angle$ LMP respectively.

To determine

To find :  The measure of $\angle$ OMQ which is formed by the angle bisectors of angles $\angle$ PMNand $\angle$ LMP by the use of a protractor.

  

Answer to Problem 12WE

The measurement of angle $\angle$ OMQ is ${90}^0$ also shown below.

  

Explanation of Solution

Given information :  Angle bisectors $\overrightarrow{MQ}$ and $\overrightarrow{MO}$ bisect $\angle$ PMNand $\angle$ LMP and $\angle$ OMQ is to be measured by a protractor.

  

  Place the midpoint of the protractor on M. Align $\overrightarrow{MQ}$ with zero line of the protractor. Read the degrees where $\overrightarrow{MO}$ crosses the number scale the result will be $\angle$ OMQ = ${90}^0$ .

To determine

To prove :  The measure of $\angle$ OMQwhich is formed by the angle bisectors of angles $\angle$ PMNand $\angle$ LMP is ${90}^0$ without the use of a protractor.

  

Explanation of Solution

Given information :  Angle bisectors $\overrightarrow{MQ}$ and $\overrightarrow{MO}$ bisect $\angle$ PMNand $\angle$ LMPrespectively .

  

Formula used :  Angles on a straight line sum up to ${180}^0$ .

An angle bisector is a line which cuts or divides an angle into two equal parts or angles.

Proof :  Angles on a straight line sum up to ${180}^0$ , so

  $\begin{array}{l}m\angle PMN+m\angle LMP=\text{ 18}{0}^0\\ m\angle PMQ+m\angle QMN+m\angle LMO+m\angle OMP=\text{ 18}{0}^0\end{array}$

Now, $\overrightarrow{MQ}$ and $\overrightarrow{MO}$ bisect $\angle$ PMNand $\angle$ LMP respectively ,

So $m\angle PMQ=m\angle QMN$ ; and $m\angle LMO=m\angle OMP$ .

So, $m\angle PMQ+m\angle QMN+m\angle LMO+m\angle OMP={180}^0$ becomes

  $\begin{array}{l}2m\angle PMQ+\text{2}m\angle OMP={180}^0\\ m\angle PMQ+m\angle OMP={90}^0\\ m\angle OMQ={90}^0\end{array}$

Hence it can be proved without a protractor that $m\angle OMQ={90}^0$ .