Construct the angle of the following measurement 30 degree

[ fig. is in the 3 attachments.....]

Solution:

(i) 30°

Steps of Construction:

1.Taking O as centre and
some radius , draw an arc of a circle, which intersects OA, say at a point B.

2.Taking B as centre and
with the same radius as before, draw an arc intersecting the previously drawn
arc, say at a point C.

3.Draw the ray OE
passing through C. Then ∠EOA = 60°.

4. Taking B and C as
centres and with the radius more than 1/2 BC, draw arcs to intersect each other,  at D.

5. Draw the ray OD. This
ray OD is the bisector of the ∠ EOA, i.e., 

∠EOD = ∠AOD =1/2 ∠EOA = 1/2(60°) = 30∘.

∠AOD =30∘.

(ii) 22½°

Steps of Construction:

1. Taking O as centre and
some radius, draw an arc of a circle, which intersects OA, at a point B.

2.Taking B as centre and
with the same radius as before, draw an arc intersecting the previously drawn
arc,at a point C .

3.Taking C as centre and
with the same radius as before, draw an arc intersecting the arc drawn in step
1, at D.

4. Draw the ray OE
passing through C. Then ∠EOA = 60∘.

5. Draw the ray OF
passing through D. Then ∠ FOE = 60∘.

6. Next, taking C and D
as centres and with radius more than 1/2 CD, draw arcs to intersect each other, say at G.

7. Draw the ray OG. This
ray OG is the bisector of the ∠ FOE, i.e., 

∠FOG = ∠EOG = 1/2 ∠FOE = 1/2 (60∘) = 30∘.

Thus, ∠GOA = ∠GOE + ∠EOA = 30∘ + 60∘ = 90∘.

8.  Now, taking O as
centre and any radius, draw an arc to intersect the rays OA and OG, say at H
and I.

9. Next, taking H and I
as centres and with the radius more than 1/2HI, draw arcs to intersect each other, at J.

10.  Draw the ray OJ. This ray OJ is the bisector of the∠ GOA.
i.e., 

∠GOJ = ∠AOJ = 1/2 ∠GOA = 1/2( 90∘) = 45∘.

11.  Now, taking O as centre and any radius, draw an arc to intersect
the rays OA and OJ, say at K and L.

12.  Next, taking K and L as centres and with the radius more
than 1/2KL, draw arcs to
intersect each other, at M.

13.  Draw the ray OM. This ray OM is the bisector of the angle AOJ,
i.e., ∠JOM = ∠AOM = 1/2 ∠AOJ = 1/2( 45∘ ) = 22½∘

 ∠AOM = 22½∘

(iii) 15∘

Steps of construction:

1.  Taking O as centre and
some radius, draw an arc of a circle, which intersects OA, at a point B.

2. Taking B as centre and
with the same radius as before, draw an arc intersecting the previously drawn
arc,at a point C.

3. Draw the ray OE
passing through C. Then ∠EOA =60∘.

4.Now, taking B and C as
centres and with the radius more than 1/2BC, draw arcs to intersect each other, at D.

5. Draw the ray OD
intersecting the arc drawn in step 1 at F. This ray OD is the bisector of the
angle EOA,

i.e., ∠EOD = ∠AOD = 1/2 ∠EOA = 1/2 (60∘) = 30∘.

6. Now, taking B and F as
centres and with the radius more than 1/2 BF, draw arcs to intersect each other, at G.

7. Draw the ray OG. This
ray OG is the bisector of the∠AOD, i.e., 

∠DOG = ∠AOG = 1/2 ∠AOD = 1/2 (30∘) = 15∘.

∠AOG = 15∘.

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Hope this will help you....