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Construct an angle of 45° at the initial point of a given ray and justify the construction.

Given: A ray OA.
Required: To construct an angle of 45° at O and justify the construction.
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. Taking C as centre and with the same radius as before, draw an arc intersecting the arc drawn in step 1, say at D.


Given: A ray OA.Required: To construct an angle of 45° at O and just

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 half CD, draw arcs to intersect each other, say at G.
7.  Draw the ray OG. This ray OG is the bisector of the angle FOE, i.e., ∠FOG

equals angle EOG equals 1 half angle FOE equals 1 half left parenthesis 60 degree right parenthesis equals 30 degree
Thus comma space space angle GOA space equals space angle GOE space plus space angle EOA space equals space 30 degree plus 60 degree equals 90 degree

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

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

10. Draw the ray OJ. This ray OJ is the required bisector of the angle GOA.

Thus comma space space space angle GOJ equals angle AOJ equals 1 half angle GOA
space space space space space space space space space space space space space space space space space space space equals space 1 half left parenthesis 90 degree right parenthesis equals 45 degree

Justification:
(i) Join BC.
Then, OC = OB = BC (By construction)
∴ ∆COB is an equilateral triangle.
∴ ∠COB = 60°.
∴ ∠EOA = 60°.
(ii) Join CD.
Then, OD = OC = CD (By construction)
∴ ∆DOC is an equilateral triangle.
∴ ∠DOC = 60°.
∴ ∠FOE = 60°.
(iii) Join CG and DG.
In ∆ODG and ∆OCG,

OD = OC                    | Radi fo the same arc
DG = CG                       | Arcs of equal radii
OG = OG                                   | Common

therefore space space space space increment ODG space equals space increment OCG space space space space space space space space space space space space space space space space space vertical line space SSS space Rule
therefore space space space space space angle DOG space equals space angle COG space space space space space space space space space space space space space space space space space space space space space vertical line thin space CPCT
therefore space space angle FOG equals angle EOG equals 1 half angle FOE
space space space space space space space space space space space space space space space equals 1 half left parenthesis 60 degree right parenthesis equals 30 degree
Thus comma space angle GOA equals angle GOE plus angle EOA
space space space space space space space space space space space space space space space space space space space equals space 30 degree plus 60 degree equals 90 degree

(iv) Join HJ and IJ

In space increment OIJ space space and space increment OHJ

OI = OH                | Radii of the same arc
IJ = HJ                    | Arcs of equal radii
OJ=OJ                                | Common

therefore space space space increment OIJ thin space equals space increment OHJ space space space space space space space space space space space space space space space vertical line thin space SSS space Rule
therefore space space space angle IOJ space equals space angle HOJ space space space space space space space space space space space space space space space space space space space vertical line space CPCT
therefore space space space angle AOJ equals angle GOJ equals 1 half angle GOA
space space space space space space space space space space space space space space space space equals space 1 half left parenthesis 90 degree right parenthesis equals 45 degree

886 Views

Construct the angles of the following measurements:

30 degree

(i) 30°

Given: A ray OA.
Required: To construct an angle of 30° at O.
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.


(i) 30°
Given: A ray OA.Required: To construct an angle of 30° at O

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 half BC, draw arcs to intersect each other, say at D.
5. Draw the ray OD. This ray OD is the bisector of the angle EOA, i.e.,

          angle EOD equals angle AOD equals 1 half angle EOA
space space space space space space space space space space equals space 1 half left parenthesis 60 degree right parenthesis equals 30 degree

582 Views

Construct the angles of the following measurements:
22 1 half degree

22 1 half degree

Given : A rayOA
Required : To construct an angle of 22 1 half degree at 0.

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.


Given : A rayOARequired : To construct an angle of  at 0.Steps of C

3. Taking C as centre and with the same radius as before, draw an arc intersecting the arc drawn in step 1, say 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 half CD, draw arcs to intersect each other, say at G.
7. Draw the ray OG. This ray OG is the bisector of the angle FOE, i.e.,

angle FOG equals angle EOG equals 1 half angle FOE
space space space space space space space space space equals space 1 half left parenthesis 60 degree right parenthesis space equals space 30 degree
Thus comma space angle GOA equals angle GOE plus angle EOA
space space space space space space space space space space space space space space space space space space equals space 30 degree plus 60 degree equals 90 degree

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

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

10. Draw the ray OJ. This ray OJ is the bisector of the angle GOA.

straight i. straight e. comma space space angle GOJ equals angle AOJ equals 1 half angle GOA
space space space space space space space space space space space space space space space space space equals space 1 half left parenthesis 90 degree right parenthesis equals 45 degree

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

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

13. Draw the ray OM. This ray OM is the bisector of the angle AOJ, i.e., ∠JOM = ∠AOM

equals 1 half angle AOJ equals 1 half left parenthesis 45 degree right parenthesis equals 22 1 half degree

 

558 Views

Construct the angles of the following measurements:
15°

15°

Given: A ray OA.

Required: To construct an angle of 15° at O.
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.


15°
Given: A ray OA.
Required: To construct an angle of 15° at O.St

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 half BC, draw arcs to intersect each other, say 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

equals 1 half angle EOA equals 1 half left parenthesis 60 degree right parenthesis equals 30 degree

6. Now, taking B and F as centres and with the radius more than 1 half BF, draw arcs to intersect each other, say at G.
7. Draw the ray OG. This ray OG is the bisector of the angle AOD, i.e., ∠DOG =

angle AOG equals 1 half angle AOD equals 1 half left parenthesis 30 degree right parenthesis equals 15 degree.

616 Views

Construct an angle of 90° at the initial point of a given ray and justify the construction.


Given: A ray OA.
Required: To construct an angle of 90° at O and justify the construction.
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 are intersecting the previously drawn are, say at a point C.


Given: A ray OA.Required: To construct an angle of 90° at O and just

3. Taking C as centre and with the same radius as before, draw an arc intersecting the arc drawn in step 1, say 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 the radius more than 1 half ID, draw arcs to intersect each other, say at G.

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

equals angle EOG equals 1 half angle FOE equals 1 half left parenthesis 60 degree right parenthesis equals 30 degree
Thus comma space space space angle GOA equals angle GOE plus angle EOA equals 30 degree plus 60 degree equals 90 degree

Justification:
(i) Join BC.
Then. OC = OB = BC (By construction)
∴ ∆COB is an equilateral triangle.
∴ ∠COB = 60°.
∴ ∠EOA = 60°.
(ii) Join CD.
Then, OD = OC = CD (By construction)
∴ ∆DOC is an equilateral triangle.

therefore space space space angle DOC equals 60 degree
therefore space space space angle FOE equals 60 degree
left parenthesis iii right parenthesis space space Join space CG space and space DG
In space increment ODG space and space increment OCG comma
OD space equals OC space space space space space space space space space space space space space space vertical line space Radii space fo space the space same space arc
DG space equals space CG space space space space space space space space space space space space space space space vertical line space Arcs space of space equal space radi
OG space equals space OG space space space space space space space space space space space space space space space space space space space space space space space space space space space vertical line space Common
therefore increment ODG space approximately equal to space increment OCG space space space space space space space space space space space space space vertical line space SSS space Rule
therefore space angle DOG equals angle COG space space space space space space space space space space space space space space space space space vertical line space CPCT
therefore space space space space angle FOG equals angle EOG equals 1 half angle FOE
space space space space space space space space space space space space space space space space equals space 1 half left parenthesis 60 degree right parenthesis equals 30 degree
Thus comma space angle GOA equals angle GOE plus angle EOA
space space space space space space space space space space space space space space space space space space equals space 30 degree plus 60 degree equals 90 degree

1676 Views