Constructing angles and bisectors without protractor

To draw angles we can use a protractor. But you can do it in a different way, too, by using a compass and a ruler. Check this out! To start with, let’s draw a right angle, 90 degrees. Draw a straight line between two points.

Write capital A at one end of the line and capital B at the other end. Place the compass needle on point A. Adjust the compass so that the radius is greater than half the distance between A and B. And now, draw an arc above and below the line. Then - without changing the settings on the compass - put the needle at point B.

Do the same - draw an arc above and below the line. The new arcs should cross the first. Mark C and D where the arcs cross each other. Now: Draw a line from C to D. Write E where the new line crosses the line AB.

Since we did not change the settings on the compass, the distances between the points A and C, B and C, A and D and B and D are equal in length. Right? Because all the distances are the same as the radius of the compass! In addition, point E ends exactly half way between A and B. .. all of these things together mean that the angles at point E are exactly 90 degrees.

Sometimes we want to divide an angle into two equal parts. When we do, the line that divides the angle is called a bisector. Now we want to do it with the 90 degree angle we just drew. How do we do that? Like this: Insert the compass needle at point E.

Draw an arc that crosses the lines AB and CD. Keep the same radius on the compass - and place the needle where the arc crosses CD. Then draw another arc, between the lines AB and CD. Continue with the same radius on the compass. Draw another arc, now with the compass needle where the arc crosses AB.

And draw a line from E to the point where the arcs cross each other. Since the radius of the compass did not change, the arcs cross at exactly the same distance from points C and B. And that means that if we draw a line from that point to point E, the right angle at E will be divided into two equal parts. That is, two angles of 45 degrees. Without using a protractor we have drawn a 90-degree angle and divided it into two 45-degree angles.

Now let’s draw angles that are 60 degrees. In an equilateral triangle, all the angles are 60 degrees. We can use this to help us. Take a compass and a ruler. First, draw a line between two points.

Mark one endpoint P, and the other Q. Place the needle at P and draw an arc that goes to Q. The radius then has the same length as the line PQ. Do the same with the needle on Q and draw an arc that goes to P. Mark as R, the point, where the arcs cross each other.

And now, draw a line between P and R and between Q and R. The radius of the compass was the same length as it was when we drew the lines PQ and QR. Now we have a triangle with three equal sides - an equilateral triangle - and therefore the angle between the lines is 60 degrees. To draw angles with other degrees, we can combine 60-, 90- or 45-degree angles. 135 degrees is 90 plus 45.

150 degrees is 90 plus 60 15 degrees is 60 minus 45.. And you can keep on and on..