question_answer 29)

Construct with ruler and compasses, angle of the following measures. (a) \[{{60}^{o}}\]                                              (b) \[{{30}^{o}}\] (c) \[{{90}^{o}}\]                                              (d) \[{{120}^{o}}\] (e) \[{{45}^{o}}\]                                              (f) \[{{135}^{o}}\]

Answer:

                (a) For constructing an angle of \[{{60}^{o}}\], we use the following steps of construction: Step I Draw a line l and mark a point O on it. Step II Place the pointer of the compasses at O and draw an arc of convenient radius which cuts the line I at a point say A. Step III With the pointer at A (as centre), now draw an arc that passes through O. Step IV Let the two arcs intersect at B. Join OB. Then, we get \[{{l}_{1}},{{l}_{2}},{{l}_{3}},{{l}_{4}},{{l}_{5}}\] whose measure is \[{{60}^{o}}\]. (b) For constructing an angle of \[{{30}^{o}}\], we firstly construct an angle of \[{{60}^{o}}\] and then bisect it. Here, we use the following steps of construction: Step I   Draw a line I and mark a point O on it. Step II   Place the pointer of the compasses at O and draw an arc of convenient radius which cuts the line \[l\] at a point, say A. Step III With the pointer at A (as centre), draw an arc that passes through O. Step IV Let the two arcs intersect at B. Join OB. Then, we get \[{{l}_{6}}\]whose measure is 60. Step V With A as centre, draw (in the interior of \[{{l}_{1}},{{l}_{2}},{{l}_{3}},{{l}_{4}}\] an arc, whose radius is more than half the length of AB). Step VI With the same radius and with B as centre, draw another arc in the interior of \[{{l}_{5}}\]. Let the two arcs intersect at D. Join OD. Then, \[{{l}_{1}},{{l}_{2}}\] is the bisector of \[{{l}_{3}}\] Thus, we get \[{{l}_{1}},{{l}_{2}},{{l}_{3}}\]and \[{{l}_{4}}\]which are equal in measure. On measuring, \[{{l}_{1}}\] (c) To draw an angle of measure \[{{90}^{o}}\], we use following steps of construction: Step I   Draw a line I and mark point O and A on it. Step II   Place the pointer of the compasses at O and draw an arc of convenient radius which cuts the line I at a point P. Step III Without disturbing the radius on the compasses, draw an arc with P as centre which cuts the first arc at Q. Step IV Again, without disturbing the radius on the compasses and with Q as centre, draw an arc which cuts the arc (drawn in Step II) at R. Step V   Now, with Q as centre and with radius more than half of length RQ draw an arc. Step VI Without disturbing the radius on the compasses, draw another arc with R as centre, which cuts the arc draw in Step V at B. Step VII Join OB. Then, we get \[{{l}_{2}}\] which is of measure \[{{90}^{o}}\]. (d) To draw an angle of measure \[{{120}^{o}}\], use the following steps of construction : An angle of \[{{120}^{o}}\] is nothing but twice of an angle of \[{{60}^{o}}\]. Therefore, it can be constructed by using the following steps of construction. Step I Draw any line PQ and take a point O on it. Step II   Place the pointer of the compasses at O and draw an arc of convenient radius which cuts the line at A. Step III Without disturbing the radius on the compasses, draw an arc with A as centre, which cuts the first arc at B. Step IV Again, without disturbing the radius on the compasses and with B as centre, draw an arc which cuts the first arc (drawn in Step II) at C. Step V   Join OC, Thus, \[{{l}_{1}}\]is the required angle, whose measure is \[{{120}^{o}}\]. (e) To draw an angle of measure \[{{45}^{o}}\], we use the following steps of construction: Step I   Draw a line I and mark points O and A on it. Step II   Place the pointer of the compasses of O and draw an arc of convenient radius, which cuts the line I at a point P. Step III Without disturbing the radius on the compasses, draw an arc with P as centre, which cuts the first arc at Q. Step IV Again without disturbing the radius on the compasses and with Q as centre, draw an arc, which cuts the first arc (drawn in Step II) at E. Step V Now, with Q as centre and with radius more than half of length RQ draw an arc. Step VI Without disturbing the radius on the compasses, draw another arc with R as centre, which cut the arc drawn in Step V at B. Step VII Join OB, let it cut the arc QR at I. Then, we get \[{{l}_{1}}\] which is of measure\[{{90}^{o}}\]. Step VIII Now, with P as centre and with radius more than half of length PI, draw an arc. Step IX Without disturbing the radius on the compasses, draw another arc with I as centre which cuts the arc drawn in Step VIII at C. Step X   Join OC. Then, we get \[{{l}_{2}}\]which is the required angle of measure \[{{45}^{o}}\]. (f) To construct an angle of measure \[{{135}^{o}}\], we use the following steps of construction: TIPS We know that, \[S\to \]\[Y\to \] So, firstly construct an angle of \[{{120}^{o}}\] and then angle of \[{{150}^{o}}\]. Then, bisect the angle between \[{{120}^{o}}\] and \[{{150}^{o}}\] to get required angle of measure \[{{135}^{o}}\]. Step I   Draw any line PQ and take a point O on it. Step II   Place the pointer of the compasses at O and draw an arc of convenient radius which cuts the line at A and D. Step III Without disturbing the radius of the compasses, draw an arc with A as centre, which cuts the first arc at B. Step IV Again, with the same radius draw an arc with B as centre which cuts the first arc (as drawn in Step II) at C. Join OC by dotted line. Then, we get \[M\to \] Step V Now, with C and D as centre and radius more than half of length CD, draw arcs which cut each other at E. Join OE by dotted line, we get \[E\to \] Also, OE cuts the arc CD at 7. Step VI With C and l as centre and radius more than half of length CI, draw arcs which cut each other at F. Join OF. Then, we get \[T\to \]