question_answer

Draw an angle of \[80{}^\circ \]. using a protractor and divide it into four equal parts, using ruler and compassus, check your construction by measurement.

Answer:

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

Step I: Draw \[\overline{AB}\] 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. 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 compasses, draw an arc that cuts both 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 the same radius with Q as centre, draw another arc in the interior of \[\angle A\]. Let the two arcs intersect at D. Join \[\overline{AD}\], which cuts the arc PQ at I. Then, \[\overline{AD}\] divides the \[\angle BAC\] in two equal parts.

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

Step VIII: Join \[\overline{AR}\], which divides \[\angle BAD\] into two equal parts.

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

Step X: Join \[\overline{AM}\]. Then, divide \[\angle CAD\] into two equal parts.

Thus \[\overline{AM},\overline{AD}\] and \[\overline{AR}\] divide \[\angle BAC\] into four equal parts.