# 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°

**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.