Constructing a 45-degree Angle

Date: 06/02/98 at 15:15:23
From: Tiger Lily
Subject: Constructing a 45-degree angle without a protractor

Dr. Math:

This is probably an easy question but I can't find an explanation in 
my math book. How do you construct a 45-degree angle with only a 
compass and a straightedge?  I know how to copy and bisect angles with 
only a compass and a straightedge. But how do I actually make a 
specific degree angle?  

Please help.
Tiger Lily

Date: 06/15/98 at 19:34:16
From: Doctor Jeremiah
Subject: Re: Constructing a 45-degree angle without a protractor

Hi. To create an angle of 45 degrees we need two lines that intersect 
at a 45-degree angle.

One way of doing this is to create a 90-degree angle and bisect it. 
This requires only a compass and straight edge.

To create a 90-degree angle we need an original line that has 
another line perpendicular to it. This requires only a compass and 
straight edge.

To create a perpendicular line we need to create two points to draw 
the line through.

Step 1:  Draw the original line

   --------A-----------------B----------

Step 2:  Mark temporary arcs from point A

                   \
                    \
                     \




   --------A-----------------B----------





                     /
                    /
                   /


Step 3:  Mark temporary arcs from point B

                   \ /
                    X
                   / \





   --------A-----------------B----------
 




                   \ /
                    X
                   / \


Step 4: Draw a line through the points

                   \|/
                    X
                   /|\
                    |
                    |
                    |
                    |
                    |90
   --------A--------+--------B----------
                    |
                    |
                    |
                    |
                    |
                   \|/
                    X
                   /|\

Now we have the right angle we need to bisect to make a 45-degree 
angle. The only difference between bisecting a 45-degree angle and 
what we just did is that we don't need to draw two sets of marks.

We only need to draw one set of marks because the intersection between 
the original line and the perpendicular line is the other.

So now we draw a line through the cross we just drew and the 
intersection and we get:

   |
   |             +
   |         | +
   C       --+--
   |       + |
   |     +
   |   +
   | + 45
   +---------B----------

-Doctor Jeremiah,  The Math Forum
Check out our web site! http://mathforum.org/dr.math/