# Construct the following angles and verify by measuring them by a protractor:

Question. Construct the following angles and verify by measuring them by a protractor:

(i) $75^{\circ}$

(ii) $105^{\circ}$

(iii) $135^{\circ}$

Solution:

(i) $75^{\circ}$

The below given steps will be followed to construct an angle of 75°.

(1) Take the given ray PQ. Draw an arc of some radius taking point P as its centre, which intersects PQ at R.

(2) Taking R as centre and with the same radius as before, draw an arc intersecting the previously drawn arc at S.

(3) Taking S as centre and with the same radius as before, draw an arc intersecting the arc at T (see figure).

(4) Taking S and T as centre, draw an arc of same radius to intersect each other at U.

(5) Join PU. Let it intersect the arc at $V$. Taking $S$ and $V$ as centre, draw arcs with radius more than $\frac{1}{2} S V$. Let those intersect each other at $W$.Join PW which is the required ray making 75° with the given ray PQ.

The angle so formed can be measured with the help of a protractor. It comes to be 75º.

(ii) $105^{\circ}$

The below given steps will be followed to construct an angle of 105°.

(1) Take the given ray PQ. Draw an arc of some radius taking point P as its centre, which intersects PQ at R.

(2) Taking R as centre and with the same radius as before, draw an arc intersecting the previously drawn arc at S.

(3) Taking S as centre and with the same radius as before, draw an arc intersecting the arc at T (see figure).

(4) Taking S and T as centre, draw an arc of same radius to intersect each other at U.

(5) Join PU. Let it intersect the arc at $V$. Taking $T$ and $V$ as centre, draw arcs with radius more than $\frac{1}{2} T V$. Let these arcs intersect each other at W. Join PW which is the required ray making 105° with the given ray PQ.

The angle so formed can be measured with the help of a protractor. It comes to be 105º.

(iii) $135^{\circ}$

The below given steps will be followed to construct an angle of 135°.

(1) Take the given ray PQ. Extend PQ on the opposite side of Q. Draw a semi-circle of some radius taking point P as its centre, which intersects PQ at R and W.

(2) Taking R as centre and with the same radius as before, draw an arc intersecting the previously drawn arc at S.

(3) Taking S as centre and with the same radius as before, draw an arc intersecting the arc at T (see figure).

(4) Taking S and T as centre, draw an arc of same radius to intersect each other at U.

(5) Join PU. Let it intersect the arc at $V$. Taking $V$ and $W$ as centre and with radius more than $\frac{1}{2} V W$, draw arcs to intersect each other at $X$. Join $\mathrm{PX}$, which is the required ray making $135^{\circ}$ with the given line $\mathrm{PQ}$.

The angle so formed can be measured with the help of a protractor. It comes to be 135º.