Question

Construct the following angles with the help of ruler and compasses only:

(i) 30°
(ii) 90°
(iii) 45°
(iv) 135°
(v) 150°
(vi) 105°

Answer 1

(i) 30°
Draw a ray OA.
With a convenient radius and centre at O, draw an arc, which cuts OA at P.
With the same radius and centre at P, draw an arc cutting the previous arc at Q.
Taking P and Q as centres and radius more than half of PQ, draw two arcs, which cut each other at R.
Draw OR and extend it to B.
$\angle$AOB is the required angle of 30°.

(ii) 90°
Draw a ray OA.
With a convenient radius and centre at O, draw an arc cutting the ray OA at P.
With the same radius ​and centre at P, draw another arc, which cuts the first arc at Q.
With the same radius ​and centre at Q, draw another arc, which cuts the first arc at R.
With Q and R as centres and radius more than half of QR, draw two arcs, which cut each other at S.
Draw OS and extend it to B to form the ray OB.
$\angle$AOB is the required angle of 90°.

(iii) 45°
To construct an angle of 45°, construct an angle of 90° and bisect it.
Construct the angle $\angle$AOB = 90°, where rays OA and OB intersect the arc at points P and T as shown in figure.
With P and T as centres and radius more than half of PT, draw two arcs, which cut each other at X.
Draw OX and extend it to C to form the ray OC.
$\angle$AOC is the required angle of 45°.

(iv) 135°
Draw a line AB and take a point O at the middle of AB.
With a convenient radius and centre at O, draw an arc, which cuts AB at P and Q, respectively.
Draw an angle of 90° on the ray OB as $\angle$BOC = 90°, where ray OC cuts the arc at R.
With Q and R as centres and radius more than half of QR, draw two arcs, which cut each other at S.
Draw OS and extend it to D to form the ray OD.
$\angle$BOD is required angle of 135°.

(v) 150°
Draw a line AB and take a point O at the middle of AB.
With a convenient radius and centre at O, draw an arc, which cuts the line AB at P and Q.
With the same radius and centre at Q, draw an arc, which cuts the first arc at R.
With the same radius and centre at R , draw an arc which cuts the first arc at S.
With centres P and S and radius more than half of PS, draw two arcs, which cut each other at T.
Draw OT and extend it to C to form the ray OC.
$\angle$BOC is required angle of 150°.

(vi) 105°
Draw a ray OA and make an angle $\angle$AOB = 90° and $\angle$AOC = 120°
Now, bisect $\angle$BOC and gets the ray OD.
$\angle$AOD is the required angle of 105°.