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Construct an angle of \( 45^{\circ} \) at the initial point of a given ray and justify the construction.
To do:
We have to construct an angle of $45^o$ at the initial point of a given ray and justify the construction.
Solution:
Steps of construction:
(a) Draw a ray $BC$.
(b) With center $B$ and a suitable radius draw an arc meeting $BC$ at $E$.
(c) With center $E$ and the same radius as above draw an arc intersecting the above arc at $F$
(d) With center $F$ and the same radius as above draw an arc intersecting the above arc at $G$
(e) With $E$ and $F$ as centers and radius more than $\frac{1}{2}EF$, draw two arcs meeting each other at $H$.
(f) Join $BH$ and produce it to $X$ so that $\angle XBC = 90^o$.
(g) Let $BH$ intersects the first arc at $D$
(h) With $D$ and $E$ as centers and radius greater than $\frac{1}{2}DE$, draw two arcs meeting each other at $K$
(i) Join $BK$ and extend it to $A$ forming ray $BA$
(j) Therefore, $\angle CBA=45^o$
Justification:
We have,
$\angle HBC=90^o$
We also have a perpendicular bisector $BA$ which divides $\angle HBC$
This implies,
$ABC=\frac{1}{2}HBC$
$ABC=\frac{1}{2}(90^o)$
$ABC=45^o$
Therefore, justified.