91Ó°ÊÓ

The figure constitutes of a straight-line LN and the line PM intersect LN at M.

Draw a line and mark points L and Nas vertex. Mark a point Msomewhere on the line LN. Take a protractor and place its horizontal edge on LNand center on M. Measure $∠PMN$from the given figure. Start from zero on the right, read the scale on the protractor and mark the reading equal to $∠PMN$. Mark the point P and join $\stackrel{→}{MP}$.

Therefore, the resulting figure is same as the one given.

The angles $∠LMP$and$∠PMN$ are shown in the figure below,

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

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

Therefore, $\stackrel{→}{MQ}$and $\stackrel{→}{MO}$ are the angle bisectors of angles$∠PMN$ and$∠LMP$ respectively.

$\stackrel{→}{MQ}$and $\stackrel{→}{MO}$ are the angle bisectors of angles$∠PMN$ and$∠LMP$ respectively.

Here, the angle$∠OMQ$ is to be measured.

Place the midpoint of the protractor on M. Align$\stackrel{→}{MQ}$ with zero line of the protractor. Read the degrees where $\stackrel{→}{MO}$ crosses the number scale.

The measure of the angle formed by the bisectors is:

$∠OMQ={90}^{∘}$.

The diagram representing it is:

$\stackrel{→}{MQ}$and $\stackrel{→}{MO}$ are the angle bisectors of angles$∠PMN$ and$∠LMP$ respectively.

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

The formula to be used is angles in a straight line sum up to ${180}^{∘}$.

Proof:

$\begin{array}{l}m∠PMN+m∠LMP={180}^{∘}\\ m∠PMQ+m∠QMN+m∠LMO+m∠OMP={180}^{∘}\end{array}$

Now, $\stackrel{→}{MQ}$and $\stackrel{→}{MO}$are the angle bisectors of angles $∠PMN$ and $∠LMP$respectively.

So, $m∠PMQ=m∠QMN$and $m∠LMO=m∠OMP$.

So, $m∠PMQ+m∠QMN+m∠LMO+m∠OMP={180}^{∘}$becomes,

$$\begin{gathered} 2m∠PMQ+2m∠OMP={180}^{∘} \\ m∠PMQ+m∠OMP={90}^{∘} \\ m∠OMQ={90}^{∘} \end{gathered}$$

Therefore, the measure of the angle formed by the bisectors is ${90}^{∘}$ can be measured without a protractor by using the formula angles in a straight line sum up to ${180}^{∘}$.