Question

If I do the following constructions step by step, what will be the angles constructed (in degrees)?

Step 1. Draw an arc on the line XY with an arbitrary point O on line XY, as centre and radius r. With the same radius and point of intersection of arc and line, ‘A’, as centre draw another arc on the first arc at point B. So, $\angle AOB$ is __.

Answer 1

Statement: Draw an arc on the line XY with an arbitrary point O on line XY, as
centre and radius r. With the same radius and point of intersection of arc and line, ‘A’, as centre, draw another arc on the first arc at point B. The $\angle$AOB is 60${}^{\circ }$.
Reason: When you divide a circle into 6 equal parts, each sector will form 60${}^{\circ }$ at the centre. Consider $△$AOB in the circle. $\angle$AOB is 60${}^{\circ }$ and AO = BO = radius of the circle, we can assume that the $△$POQ is isosceles, and $\angle$ABO = $\angle$BAO = 60${}^{\circ }$.

SSince all three triangles are 60${}^{\circ }$, $△$AOB is actually equilateral and side AB is also equal to radius. Therefore, when you cut the circumference of a circle with the radius of the circle itself, the two ends of the arc forms 60${}^{\circ }$ at the centre. $\angle$AOB is hence 60${}^{\circ }$.