Construct the following angles and verify by measuring them by a protractor:
(i) 75° (ii) 105° (iii) 135°
Solution:
(i) 75°
We need to construct two adjacent angles of 60°. The second angle should be bisected twice to get a 15° angle.
75° = 60° + 15°
15° = 30°/2 = (60/2) ÷ 2
Steps of Construction:
a) Draw a ray PQ.
b) To construct an angle of 60°, with P as a center and any radius, draw a wide arc to intersect PQ at R. With R as a center and same radius, draw an arc to intersect the initial arc at S. Thus,∠SPR = 60°
(iii) To construct an adjacent angle of 60° with S as a center and the same radius, draw an arc that intersects the initial arc at T.
(iv) To bisect ∠SPT, with T and S as centers and radius more than half of TS draw arcs to bisect each other at U. Join U and P.
∠UPS = 1/2 ∠TPS = 1/2 × 60° = 30°
(v) To bisect ∠UPS, with V and S as centers and radius greater than half of VS draw arcs to intersect each other at X.
∠XPS = 1/2 ∠UPS = 1/2 × 30° =15°
∠XPQ = ∠XPS + ∠SPQ
= 15° + 60°
= 75°
Thus, ∠XPQ = 75°
(ii) 105°
We need to construct two adjacent angles of 60°. In the second angle, we need to bisect it to get two 30° angles. The second 30° angle should be bisected again to get a 15° angle. Together we can make an angle of 105°.
105° = 60° + 45°
105°= 60° + 30° + 15°
Steps of Construction:
a) Draw a ray PQ.
b)To construct an angle of 60° With P as a center and any radius, draw a wide arc to intersect PQ at R. With R as a center and same radius, draw an arc to intersect the initial arc at S. Thus, ∠SPR = 60°.
c)To construct an adjacent angle with S as the center and the same radius as before, draw an arc to intersect the initial arc at T. ∠TPS = 60°.
d) To bisect ∠TPS, with T and S as centers and radius greater than TS draw arcs to bisect each other at U. Join U and P.
∠UPS = 1/2 ∠TPS = 1/2 × 60° = 30°
∠UPT = ∠UPS = 30°
e) To bisect ∠UPT, with T and V as centers and radius greater than half of TV, draw arcs to intersect each other at W. Join P and W.
∠WPU = 1/2 ∠UPT = 1/2 × 30° = 15°
∠WPR = ∠WPU + ∠UPS + ∠SPR
= 15° + 30° + 60°
= 105°
(iii) 135°
We need to construct three adjacent angles of 60° each. The third angle should be bisected twice successively to get an angle of 15°. Together we will get an angle of 135°.
135° = 15° + 60° + 60°
15° = (60°/2) ÷ 2
Steps of Construction:
a) Draw a ray PQ.
b) To construct an angle of 60°, with P as a center and any radius, draw a wide arc to intersect PQ at R. Thus, ∠SPR = 60°.
(iii) To construct an adjacent angle of 60°, with S as the center and the same radius as before, draw an arc to intersect the initial arc at T. Thus, ∠TPS = 60°.
(iv) To construct the second adjacent angle of 60°, with T as a center and the same radius as before, draw an arc to intersect the initial arc at U. Thus, ∠UPT = 60°
(v) To bisect ∠UPT, with T and U as centers and radius greater than half of TU, draw two arcs to intersect each other at V.
∠VPT = ∠VPU = 1/2 ∠UPT = 1/2 × 60° = 30°
(vi) To bisect ∠VPT, with W and T as centers and radius greater than half of WT, draw arcs to intersect each other at X.
∠XPT = ∠XPV = 1/2 ∠VPT = 1/2 × 30° = 15°
∠XPQ = ∠XPT + ∠TPS + ∠SPR
= 15° + 60° + 60°
= 135°
Thus, ∠XPQ = 135°
☛ Check: NCERT Solutions Class 9 Maths Chapter 11
Video Solution:
Construct the following angles and verify by measuring them by a protractor: (i) 75° (ii) 105° (iii) 135°
Maths NCERT Solutions Class 9 Chapter 11 Exercise 11.1 Question 4
Summary:
It is given that we have to construct an angle of 75°, 105°, and 135° at the initial point of a given ray. We have drawn the angle using a compass and ruler and justified the construction.
☛ Related Questions:
- Construct an angle of 90° at the initial point of a given ray and justify the construction.
- Construct an angle of 45° at the initial point of a given ray and justify the construction.
- Construct the angles of the following measurements:(i) 30°(ii) 22(1/2)°(iii) 15°
- Construct an equilateral triangle, given its side and justify the construction.