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Construct a triangle given two angles and the included side (ASA)
Click here for a printable ASA triangle construction worksheet

How to construct a triangle when given the length of one side and the measures of the angles on each end of that side, using only a compass and straightedge. This is sometimes called an ASA triangle construction (angle-side-angle).

We start with a given side length, and two angles. The result is the triangle ABC whose side and two adjacent angle measures are those desired.

Instructions Click on 'Next' to go through the construction one step at a time, or click on 'Run' to let it run without stopping.
(If there is no image below, see support page.)


The two given angles are only there to indicate the measure of the two angles. The lines making up the given angles have random lengths that have no significance in the construction.

Note:  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.

Another way to think about it: The interior angles of a triangle always add to 180°. If two of them already add to more than 180°, there is nothing 'left over' for the third angle. See Interior Angles of a Triangle.
Step-by-step Instructions


The first part of this construction is to copy a line segment to form one side of the new triangle. (See Copying a Line Segment).
Step 1 Mark a point A that will be one vertex of the new triangle.
Step 2 Set the compass width to the length of the segment AB.
Step 3 With the compass point on A, make an arc near the future vertex B of the triangle.
Step 4 Mark a point B on this arc. Then draw the line AB. This will be one side of the new triangle.


What we do now is essentially copy the two angles. (See Copying an Angle).
Step 5 With the compass at any convenient width, draw an arc across both lines of the given angle A.
Step 6 Without changing the compass width, draw an arc at point A on the new triangle. The arc must cross AB and also cross the future side of the triangle.
Step 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.
Step 8 Near point A draw an arc in a similar position so it crosses the arc drawn earlier. This, in effect, 'copies' the measure of the angle at P to the angle at A.
Step 9 Draw a line from A through the point where the arcs intersect. This will become the second side of the triangle. Draw it long.
Step 10 Repeat this process at B. Copying the angle measure from the given angle B to the new triangle at B. The point where the lines intersect is C, the third vertex of the triangle.
Step 10 Done. The blue triangle ABC has the side and two angle measures desired.
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.

Other constructions

Lines

Angles

Triangles

Triangle Centers

Circles, Arcs and Ellipses

Non-Euclidean constructions

(C) 2007 Copyright John Page