How to construct (draw) a triangle given two sides and the included angle

Triangle given two sides and included angle (SAS)

Geometry construction using a compass and straightedge

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.

Step-by-step Instructions

Printer friendly version

After doing this	Your work should look like this
Start with two line segments and the included angle.
1. Mark a point A that will be one vertex of the new triangle.
2. Draw a ray from point A. This will become the side AB of the new triangle, so make it longer than AB.
3. Set the compass width to the length of the given side AB.
4. Set the compass on A, and mark a point B on the ray just drawn.
5. With the compass set to any convenient width, from the point A on the given angle, draw an arc across both lines..
6. Without changing the compass width, draw a similar sized arc at point A on the new triangle.
7. Set the compass to the arc width at the given angle A. This the distance between the points where the arc intersects the sides of the angle.
8. Make a similar arc on the new triangle so it crosses the previous arc.
9. Draw a ray from A, through where the arcs intersect and onwards. This will become side AC of the triangle so make it longer than AC.
10. Set the compass width to the distance AC.
11. With the compass on A, make an arc across the second ray, creating point C.
12. Draw the line BC, the third side of the triangle
Done, the triangle ABC has the desired two side lengths and included angle.

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.

Multiple triangles possible

Proof

Constructions pages on this site

Lines

Angles

Triangles

Triangle Centers

Circles, Arcs and Ellipses

Polygons

Non-Euclidean constructions