Construct the angles of the following measurements :

(i) 30° (ii) 75° (iii) 105°

(iv) 135° (v) 15° (vi) 22

i) The steps of the required construction are:

1) Draw a line segment AB. Keeping A as the center and any radius draw a semicircle, intersecting AB at point C.

2) Keeping C as the center and radius AC, draw an arc, cutting the semicircle at point D.

3) Taking C as the center draw an arc of any radius greater than . Now, Taking D as the center and the keeping the same radius, draw another arc, intersecting the previous arc at E. Join AE.

4) ∠BAE=30°.

ii) The steps of the required construction are:

1) Draw a line segment AB. Keeping A as the center and any radius draw a semicircle, intersecting AB at point C.

2) Keeping C as the center and radius AC, draw an arc, cutting the semicircle at point D. Keeping D as the center and radius AC, draw an arc, cutting the semicircle at point E.

3) Taking D as the center draw an arc of any radius greater than . Now, Taking E as the center and the keeping the same radius, draw another arc, intersecting the previous arc at F. Join AF, intersecting the semi-circle at G.

4) Taking D as the center draw an arc of any radius greater than . Now, Taking G as the center and the keeping the same radius, draw another arc, intersecting the previous arc at H. Join AH.

5) ∠BAH=75°.

iii) The steps of the required construction are:

1) Draw a line segment AB. Keeping A as the center and any radius draw a semicircle, intersecting AB at point C.

2) Keeping C as the center and radius AC, draw an arc, cutting the semicircle at point D. Keeping D as the center and radius AC, draw an arc, cutting the semicircle at point E.

3) Taking D as the center draw an arc of any radius greater than . Now, Taking E as the center and the keeping the same radius, draw another arc, intersecting the previous arc at F. Join AF, intersecting the semi-circle at G.

4) Taking E as the center draw an arc of any radius greater than . Now, Taking G as the center and the keeping the same radius, draw another arc, intersecting the previous arc at H. Join AH.

5) ∠BAH=105°.

iv) The steps of the required construction are:

1) Draw a line segment BC. Taking any arbitrary point A on line segment BC as the center and any radius draw a semicircle, intersecting BC at points D and E.

2) Taking D as the center draw an arc of any radius greater than . Now, Taking E as the center and the keeping the same radius, draw another arc, intersecting the previous arc at F. Join AF, which intersects the semicircle at point G.

3) Taking G as the center draw an arc of any radius greater than . Now, Taking E as the center and keeping the same radius, draw another arc, intersecting the previous arc at H. Join AH.

4) ∠BAF=135°.

v) The steps of the required construction are:

1) Draw a line segment AB. Keeping A as the center and any radius draw a semicircle, intersecting AB at point C.

2) Keeping C as the center and radius AC, draw an arc, cutting the semicircle at point D.

3) Taking C as the center draw an arc of any radius greater than . Now, Taking D as the center and the keeping the same radius, draw another arc, intersecting the previous arc at E. Join AE, intersecting the semi-circle at F.

4) Taking C as the center draw an arc of any radius greater than . Now, Taking F as the center and the keeping the same radius, draw another arc, intersecting the previous arc at G. Join AG.

5) ∠BAG=15°.

vi) The steps of the required construction are:

1) Draw a line segment BC. Taking any arbitrary point, A on line segment BC as the the center and any radius draw a semicircle, intersecting BC at points D and E.

2) Taking D as the center draw an arc of any radius greater than . Now, Taking E as the center and the keeping the same radius, draw another arc, intersecting the previous arc at F. Join AF, which intersects the semicircle at point G.

3) Taking D as the center draw an arc of any radius greater than . Now, Taking G as the center and the keeping the same radius, draw another arc, intersecting the previous arc at H. Join AH, which intersects the semicircle at point I.

4) Taking D as the center draw an arc of any radius greater than . Now, Taking I as the center and the keeping the same radius, draw another arc, intersecting the previous arc at J. Join AJ.

5) .