Question

Prove that the bisectors of two adjacent supplementary angles include a right angle.

Answer 1

Hint: To prove that the bisectors of two adjacent supplementary angles include a right angle we draw the supplementary angles and also construct the angle bisectors of both the angles. Then use the property of angle bisectors to get the values of these angles. Now, we realize that all the angles lie on the same line so we use the property that the sum of all angles will be equal to $180{}^{\circ }$. Now, from this, we can prove the required results.

Complete step-by-step answer:
To prove that the bisectors of two adjacent supplementary angles include a right angle we draw the supplementary angles and also construct the angle bisectors of both the angles.

In the above diagram $\mathrm{\angle }AOC$ and $\mathrm{\angle }BOC$ are supplementary angles.
In the above diagram we construct OD and OE.
OD is the angle bisector of angle $\mathrm{\angle }AOC$ and OE is the angle bisector of $\mathrm{\angle }BOC$ .
As we know that the angle bisector bisects the angle into two equal parts.
So, $\mathrm{\angle }AOD=\mathrm{\angle }COD$ and $\mathrm{\angle }BOE=\mathrm{\angle }COE$…………(i)
AB is a straight line and the sum of all angles lying on a straight line is $180{}^{\circ }$.
So, we can first equate the sum of the two supplementary angles to $180{}^{\circ }$. Then, we will get
 $\mathrm{\angle }AOC+\mathrm{\angle }BOC=180{}^{\circ }$
$\mathrm{\angle }AOD+\mathrm{\angle }COD+\mathrm{\angle }BOE+\mathrm{\angle }COE=180{}^{\circ }$ (From diagram)
Put the values from eq. (i)
$\begin{array}{rl}& \mathrm{\angle }COD+\mathrm{\angle }COD+\mathrm{\angle }COE+\mathrm{\angle }COE=180{}^{\circ }\\ & 2\mathrm{\angle }COD+2\mathrm{\angle }COE=180{}^{\circ }\\ & 2(\mathrm{\angle }COD+\mathrm{\angle }COE)=180{}^{\circ }\\ & \mathrm{\angle }COD+\mathrm{\angle }COE=90{}^{\circ }\\ & \mathrm{\angle }EOD=90{}^{\circ }\end{array}$
Hence proved
The bisectors of two adjacent supplementary angles include a right angle.

Note: In the given problem the property of straight line is used. Students must remember while drawing the supplementary angles that the sum of the measures of two angles is$180{}^{\circ }$. It is better to draw the supplementary angles on a straight line. Also students must remember the property of angle bisectors to solve the problem. Angle bisector bisects the angle into two equal parts.