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Constructing a 45° angle
Geometry construction using a compass and straightedge

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.
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Other constructions pages on this site

Lines

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Circles, Arcs and Ellipses

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Non-Euclidean constructions