Question

Construct the following angles at the initial point of a given ray and justify the construction:

(i) ${45}^{\circ }$

(ii) ${90}^{\circ }$

Answer 1

(i) ${45}^{\circ }$

Steps of construction :

(a) Draw a line segment $BC$.

(b) With centre $B$ and a suitable radius draw an arc meeting $BC$ at $E$.

(c) With centre $E$, cut off equal arcs $EF$ and $FG$.

(d) Bisect $FG$ at $H$.

(e) Join $BH$ and produce to $X$ so that $\angle XBC={90}^{\circ }$

(f) Bisect $\angle XBC\text{so that}\angle ABC={45}^{\circ }.$

(g) With $E$ as centre and radius more than half of $EG$, cut an arc. Similarly with centre as $G$ and radius more than half of $EG$ , cut an arc. Let these intersect at point $K$.

(h) Join $KB$ and extend it to point $A$. Therefore, $\angle ABC=\angle XBA={45}^{\circ }$

(ii) ${90}^{\circ }$

Steps of construction :

(a) Draw a line segment $BC$.

(b) With centre $B$ and suitable radius, draw an arc meeting $BC$ at $E$ and with centre $E$ cut off arcs from $E$ such that arc $EF=$ arc $FG$.

(c) Now bisect the arc $FG$ at $H$.

(d) Join $BH$ and produce it to $A$.

$\therefore \angle ABC={90}^{\circ }$.