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.
Printable step-by-step instructions
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.
Proof
This construction works by creating an isosceles right triangle, which is a 45-45-90 triangle. The image below is the final drawing above with the red items added.
| Argument | Reason | |
|---|---|---|
| 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°. |
- Q.E.D
Try it yourself
Click here for a printable worksheet containing two 45° angle exercises. When you get to the page, use the browser print command to print as many as you wish. The printed output is not copyright.Other constructions pages on this site
Lines
- Introduction to constructions
- Copy a line segment
- Sum of n line segments
- Difference of two line segments
- Perpendicular bisector of a line segment
- Perpendicular at a point on a line
- Perpendicular from a line through a point
- Perpendicular from endpoint of a ray
- Divide a segment into n equal parts
- Parallel line through a point (angle copy)
- Parallel line through a point (rhombus)
- Parallel line through a point (translation)
Angles
- Bisecting an angle
- Copy an angle
- Construct a 30° angle
- Construct a 45° angle
- Construct a 60° angle
- Construct a 90° angle (right angle)
- Sum of n angles
- Difference of two angles
- Supplementary angle
- Complementary angle
- Constructing 75° 105° 120° 135° 150° angles and more
Triangles
- Copy a triangle
- Isosceles triangle, given base and side
- Isosceles triangle, given base and altitude
- Isosceles triangle, given leg and apex angle
- Equilateral triangle
- 30-60-90 triangle, given the hypotenuse
- Triangle, given 3 sides (sss)
- Triangle, given one side and adjacent angles (asa)
- Triangle, given two angles and non-included side (aas)
- Triangle, given two sides and included angle (sas)
- Triangle medians
- Triangle midsegment
- Triangle altitude
- Triangle altitude (outside case)
Right triangles
- Right Triangle, given one leg and hypotenuse (HL)
- Right Triangle, given both legs (LL)
- Right Triangle, given hypotenuse and one angle (HA)
- Right Triangle, given one leg and one angle (LA)
Triangle Centers
Circles, Arcs and Ellipses
- Finding the center of a circle
- Circle given 3 points
- Tangent at a point on the circle
- Tangents through an external point
- Tangents to two circles (external)
- Tangents to two circles (internal)
- Incircle of a triangle
- Focus points of a given ellipse
- Circumcircle of a triangle
Polygons
- Square given one side
- Square inscribed in a circle
- Hexagon given one side
- Hexagon inscribed in a given circle
- Pentagon inscribed in a given circle
Non-Euclidean constructions
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