1. The Straight Line, 1. Angles, Kiselev's Geometry, Book I. Planimetry, Givental Alexander.

 Exercises.

27*. Using only compass, construct a 1° arc on a circle, if a 19° arc of this circle is given.

Solution: 

In the above figure we have a circle with center A. We have the arc BB' of 19 circular degrees. The angle 𝛂 having 19 angular degrees. We pass two lines one passing through the points AB and another passing through the points AB'. Taking the radius as AC we draw a circular arc CD which intersects the line AB' in point D. Now join the segment CD. The line CD is now a projection of the line BB', where point A is the source of light and ∠ B'AB has its sides AB' and AB as the rays forming the very ends of the light that is being emitted out of the source A. We can divide the line CD into 19 equal parts in this way: Pass a line through point D making an acute angle 𝜸 in the anti-clockwise direction with CD as its leg. Then pass a line through point C making an angle equal to 𝜸 in the same direction with DC as its leg. 

Start with point C. Put the pin of the compass onto C and the pencil point in the direction of the line passing through C and making the angle 𝜸. Keep the distance between the pin and the pencil constant and mark points in the direction of the line passing through C, with each point being marked becoming the point where the pin of the compass will be placed for the next point to be marked. Mark 19 points in such way. 

Now put the pin on the point D and in the direction of the line passing through D and making the angle equal to 𝜸, keeping the distance same mark 19 points as were marked above. 

Now we have these two lines parallel to each other and divided into 19 equal parts. Start from C. Join point C with the last marked point on the line passing through D. Then join the next marked point next to C with the marked second last point next to the last point. Establish such a correspondence. And join the points. The lines which join one point to its corresponding point will divide the segment CD into 19 equal parts. Join each of the points of division with the center point A. 

The arc CD has now been divided into 19 equal parts. Thus, its corresponding arc BB' has also been divided into 19 equal parts. Select any one of those 19 equal parts. The arc that congruent to this one part will have measure 1°. We have successfully constructed an arc of 1° on the circle to which the corresponding angle is 𝜷 with A' as its vertex.

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