Question

Draw an angle of ${80}^{\circ }$ using a protractor and divide it into four equal parts, using ruler and compass. Check your constuction by measurement.

Answer 1

Here, to divide an angle of measure ${80}^{\circ }$ into four equal parts, we use the following steps of construction:

Step I: Draw a line segment $\overline{AB}$ of any length. Place the centre of the protractor at A and the zero edge along $\overline{AB}$

Step II: Start with zero near B and mark C at ${80}^{\circ }$ .

Step III: Join AC, then$\angle BAC$ is an angle of measure ${80}^{\circ }$ .

Step IV: With A as centre and using compass, draw an arc that cuts both the rays of $\angle A$ at P and Q.

Step V: With P as centre, draw (in the interior of $\angle A$ an arc, whose radius is more than half the length of PQ.

Step VI: With Q as centre and the same radius, draw another arc in the interior of $\angle A$ Let the two arcs intersect at D. Join $\overline{AD}$ Cutting are PQ at L then, AD divides the $\angle BCA$ into two equal parts.

Step VII :Now taking P and L as centre, having radius more than half of length PL, draw two arcs respectively, which cut each other at R

Step VIII: Join $\overline{AR}$ which divides $\angle DAB$ into equal parts.

Step IX: Now taking Q and L as centre, having radius more than half of length QL draw two arcs respectively, which cut each other at M.

Step X: Join $\overline{AM}$, which divide $\angle CAD$ into two equal parts. Thus, $\overline{AM}$ $\overline{AD}$and $\overline{AR}$ divide $\angle BAC$ into four equal parts.