Construct the angles of the following measurements:
(i) 30°
(ii) 75°
(iii) 105°
(iv) 135°
(v) 15°
(vi) 22 1/2°
Answers
(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) 75°
Steps of Construction:
1) First draw a line segment AB. Taking A as the center and any radius draw an arc , intersecting AB at point C.
2) Taking C as the center and radius AC, draw an arc, cutting at point D. Taking D as the center and radius AC, draw an arc, cutting at point E.
3) Taking D as the center draw an arc of any radius greater than DE/2. Now, Taking E as the center and keeping the radius same, draw an another arc, intersecting the previous arc at F. Join AF.
4) Taking D as the center draw an arc of any radius greater than DG/2 . Now, Taking G as the center and keeping the radius same , draw an another arc, intersecting the previous arc at H. Join AH.
5) Hence, ∠BAH = 75°.
(iii) 105°
Steps of Construction:
1) First draw a line segment AB. Taking A as the center and any radius draw an arc , intersecting AB at point C.
2) Taking C as the center and radius AC, draw an arc, cutting at point D. Taking D as the center and radius AC, draw an arc, cutting at point E.
3) Taking D as the center draw an arc of any radius greater than DE/2. Now, Taking E as the center and keeping the radius same, draw an another arc, intersecting the previous arc at F. Join AF.
4) Taking E as the center draw an arc of any radius greater than GE/2 . Now, Taking G as the center and keeping the radius same , draw an another arc, intersecting the previous arc at H. Join AH.
5) Hence, ∠BAH = 105°.
(ii) 135°
Steps of Construction:
(1) First draw a line segment BC. Taking any point A on line segment BC as the center and any radius draw a semicircle.
(2) Taking D as the center draw an arc of any radius greater than DE/2. Now, Taking E as the center and keeping the radius same, draw an another arc, intersecting the previous arc at F. Join AF.
(3) Taking G as the center draw an arc of any radius greater than GE/2. Now, Taking E as the center and keeping the radius same, draw an another arc, intersecting the previous arc at H. Join AH.
(4) Hence, ∠BAF = 135°.
(v) 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∘
(vi) 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∘.
HOPE THIS ANSWER WILL HELP YOU…..
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) 75°
Steps of Construction:
1) First draw a line segment AB. Taking A as the center and any radius draw an arc , intersecting AB at point C.
2) Taking C as the center and radius AC, draw an arc, cutting at point D. Taking D as the center and radius AC, draw an arc, cutting at point E.
3) Taking D as the center draw an arc of any radius greater than DE/2. Now, Taking E as the center and keeping the radius same, draw an another arc, intersecting the previous arc at F. Join AF.
4) Taking D as the center draw an arc of any radius greater than DG/2 . Now, Taking G as the center and keeping the radius same , draw an another arc, intersecting the previous arc at H. Join AH.