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Triangle given two sides and included angle (SAS)
Geometry construction using a compass and straightedge

This page shows how to construct a triangle given two sides and the included angle with compass and straightedge or ruler. It works by first copying the angle, then copying the two line segment on to the angle. A third line completes the triangle.

Multiple triangles possible

It is possible to draw more than one triangle has the side lengths and angle measure as given. Depending on which line you start with, which end of the line you draw the angles, and whether they are above or below the line, the four triangles below are possible. All four are correct in that they satisfy the requirements, and are congruent to each other.

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

The image below is the final drawing above with the red items added.

  Argument Reason
1 Line segment MN is congruent to AB. Drawn with the same compass width. For proof see Copying a line segment
2 Line segment ML is congruent to AC. Drawn with the same compass width.
3 The angle LMN is congruent to the angle A Copied using the procedure shown in Copying an angle. See that page for the proof.
4 Triangle LNM satisfies the side lengths and angle measure given.

  - Q.E.D

Try it yourself

Click here for a printable worksheet containing two SAS triangle construction problems. 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

Angles

Triangles

Right triangles

Triangle Centers

Circles, Arcs and Ellipses

Polygons

Non-Euclidean constructions