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

**Solution:**

We need to construct two adjacent angles each of 60° and bisect the second one to construct 90°. Then, bisect the 90° angle to get 45°.

[60°^{ }+ (60°/2)] / 2 = 45°

Steps of Construction:

(i) Draw ray PQ.

(ii) To construct an angle of 60°, with P as center draw a wide arc of any radius to intersect the ray at R. With R as a center and same radius draw an arc to intersect the initial one at S. Then ∠SPR = 60°.

(iii) To construct an adjacent angle of 60°, with S as the center and same radius draw an arc to intersect the previous arc at T. Then, ∠TPS = 60°

(iv) To bisect ∠TPS, with T and S as a center and radius more than half of ST, draw arcs to intersect each other at U.

(v) Join P and U. Then, ∠UPS = 1/2 ∠TPS = 30°

∠UPQ = ∠UPS + ∠SPR

= 30° + 60°

= 90°

(vi) To bisect ∠UPQ, with R and V as centers and radius greater than half of RV, draw arc to intersect each other at W. Join PW. PW is the angle bisector of ∠UPQ.

Then, ∠WPQ = 1/2 ∠UPQ = 1/2 × 90° = 45°

So, ray PW forms an angle of 45°with ray PQ at the initial point.

**Video Solution:**

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

### Maths NCERT Solutions Class 9 - Chapter 11 Exercise 11.1 Question 2:

Summary:

It is given that we have to construct an angle of 45° at the initial point of a given ray. We have drawn the angle using a compass and ruler and justified the construction.