(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 \(\frac{CD}{2}\) .
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 \(\frac{DE}{2}\) .
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 semicircle at G.
(4) Taking D as the center draw an arc of any radius greater than \(\frac{DG}{2}\) .
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 \(\frac{DE}{2}\) .
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 \(\frac{GE}{2}\) .
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 \(\frac{DE}{2}\) .
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 \(\frac{GE}{2}\) .
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 \(\frac{CD}{2}\) .
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 semicircle at F.
(4) Taking C as the center draw an arc of any radius greater than \(\frac{CF}{2}\).
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 \(\frac{DE}{2}\).
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 \(\frac{DG}{2}\).
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 \(\frac{DI}{2}\) .
Now,
Taking I as the center and the keeping the same radius, draw another arc, intersecting the previous arc at J.
Join AJ.
(5) ∠BAJ = 22\(\frac{1}{2}\)°.
