# Ex.11.1 Q1 Constructions Solution - NCERT Maths Class 9

## Question

Construct an angle of \(90^\circ\)^{ }at the initial point of a given ray and justify the construction.

## Text Solution

**Reasoning:**

We need to construct two adjacent angles each of \(60\) degrees and bisect the second one to construct \(90\) degree.

**Steps of Construction:**

(i) Draw ray \(PQ\)

(ii) To construct \(60^{\circ}\) angle. Draw an arc of any radius with \(P\) as center intersecting \(PQ\) at \(R\) with \(R\) as center and same radius draw an arc intersecting the previous arc at \(S\). \(\begin{align}\angle {SPQ}=60^{\circ}\end{align}\)

(iii) To construct adjacent \(60^{\circ}\) angle. With \(S\) as the center and same radius as before intersecting the initial arc at \(T\) \(\angle {TPS}\) will be \(60^\circ\)

(iv) To bisect \(\angle {TPS} :\)

With \(T\) and \(S\) as centers and same radius as before draw two arcs to intersect each other at \(U\). \(\begin{align}\angle {UPS}=\frac{1}{2} \angle {TPS}=30^{\circ}\end{align}\)_{ }

(v)_{ }Join \(P\) and \(U\) to get an angle of \(60^{0}\) at initial point \(P\).

\[\begin{align} \angle {UPQ} &=\angle {UPS}+\angle {SPR} \\ &=30^{\circ}+60^{\circ} \\ &=90^{\circ} \end{align}\]