
Triangle given two sides and included angle (SAS)
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 stepbystep instructions
The above animation is available as a
printable stepbystep 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
NonEuclidean constructions
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