Question

Is it possible to construct with ruler and compasses, angle of the following measure:
${135}^o$

Enter

1-for yes, 0-for no

Answer 1

${135}^o={90}^o+{45}^o$

So, to make ${135}^o,$ we make ${90}^o$ and then ${45}^o$

$\Rightarrow$ Draw a line $OA{A}^{\prime }$

$\Rightarrow$ Taking $O$ as center and any radius, draw an arc cutting $OA$ at $B.$

$\Rightarrow$ Now, with $B$ as center and same radius as before, draw an arc intersecting the previously drawn arc at point $C.$

$\Rightarrow$ With $C$ as center and the same radius, drawn an arc cutting the arc at $D.$

$\Rightarrow$ With $C$ and $D$ as centers and radius more than $\frac{1}{2}CD,$ drawn two arcs intersecting at $P.$

Thus, $\angle AOP={90}^o$

Also, $\angle A^{\prime }OP={90}^o$

So, we bisect $\angle A^{\prime }OP$

$\Rightarrow$ Mark point $Q$ where $OP$ intersects the arc

$\Rightarrow$ With $B^{\prime }$ and $Q$ as centers and radius more than $\frac{1}{2}{B}^{\prime }Q,$ draw two arcs intersecting at $R.$

$\Rightarrow$ Join $OR$

$\therefore$ $\angle POR={45}^o$

Thus,

$\angle AOR=\angle AOP+\angle POR$

$={90}^o+{35}^o$

$={135}^o$

$\therefore$ $\angle AOR={135}^o$