Triangle given two angles and the included side (ASA)

This page shows how to construct a triangle given one side and the angle at each end of it with compass and straightedge or ruler. It works by first copying the line segment to form one side of the triangle, then copy the two angles on to each end of it to complete the triangle. As noted below, there are four possible triangles that be drawn - they are all correct.

Multiple triangles possible

It is possible to draw more than one triangle has the side length and angle measures as given. Depending on which end of the line you draw the angles, and whether they are above or below the line, four triangles are possible. All four are correct in that they satisfy the requirements, and are congruent to each other. Note: this is not always possible It is not always possible to construct a triangle from a given side length and two angles. See figure on the right. If the two given angles add to more than 180°, the sides of the triangle will diverge and never meet. See Interior Angles of a Triangle.

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 JL is congruent to AB. Drawn with the same compass width. For proof see Copying a line segment
2 The angle KJL is congruent to the angle A Copied using the procedure shown in Copying an angle. See that page for the proof.
3 The angle KLJ is congruent to the angle B Copied using the procedure shown in Copying an angle. See that page for the proof.
4 Triangle JKL satisfies the side length and two angle measure given.

- Q.E.D

Try it yourself

Click here for a printable worksheet containing two ASA 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.