Question

Which of the following angles can be constructed with the help of a ruler and a compass?

• A. ${42.5}^{\circ }$
• B. ${40}^{\circ }$
• C. ${67.5}^{\circ }$
• D. ${55}^{\circ }$

Answer 1

The correct option is C

${67.5}^{\circ }$

${42.5}^{\circ }=\frac{1}{2}\times {85}^{\circ }$ and an ${85}^{\circ }$ angle cannot be constructed with a compass and ruler only.
${40}^{\circ }=\frac{1}{2}\times {80}^{\circ }$ and an ${80}^{\circ }$ angle cannot be constructed with a compass and ruler only.
${55}^{\circ }=\frac{1}{2}\times 110$ and a ${110}^{\circ }$ angle cannot be constructed with a compass and ruler only.
However :
${67.5}^{\circ }=\frac{{135}^{\circ }}{2}=\frac{1}{2}({90}^{\circ }+{45}^{\circ })$ Since a ${135}^{\circ }$ angle can be constructed with the help of a ruler and compass, we can also construct a ${67.5}^{\circ }$ angle by bisecting the ${135}^{\circ }$ angle.