This page shows how to construct (draw) a 45 degree angle with compass and straightedge or ruler. It works by constructing an isosceles right triangle, which has interior angles of 45, 45 and 90 degrees. We use one of those 45 degree angles to get the result we need. See the proof below for more details.
The above animation is available as a printable step-by-step instruction sheet, which can be used for making handouts or when a computer is not available.
|1||Line segment AB is perpendicular to PQ.||Constructed that way. See Constructing the perpendicular bisector of a line.|
|2||Triangle APC is a right triangle||Angle ACP is 90° (from step 1)|
|3||Line segments CP,CA are congruent||Drawn with same compass width|
|4||Triangle ∆APC is isosceles.||CP = AC|
|5||Angle APC has a measure of 45°.||In isosceles triangle APC, base angles CPA and CAP are congruent. (See Isosceles Triangles). The third angle ACP is 90° and the interior angles of a triangle always add to 180. So both base angles CPA and CAP are 45°.|