This page shows how to construct a triangle given two angles and a non-included side with compasses and straightedge or ruler.
The construction has four main parts:
- The given line segment AB is copied. See Copying a line segment.
- The given angle A is copied to one end of AB. See Copying an angle.
- The given angle C is copied to be adjacent to the angle A. This in effect adds A and C.
- The third angle created at A must be the angle B because the three angles at A add to 180, so this is copied to the point B. This in turn establishes point C and the triangle is complete.
How it works
The image below is the final drawing above.
The construction involves finding the angle B in the triangle. We know where this angle has to go (on the end of AB at point B), but we must first find it's measure.
This is done by using constructions that add the given angles A and C, then subtracting the result from 180. Because the interior angles of a triangle add to 180, this must be the missing third angle B.
Proof
| Argument | Reason | |
|---|---|---|
| 1 | Triangle AB = given AB | Copied using the procedure shown in Copying a line segment. See that page for the proof. |
| 2 | m∠CAB = m∠A | Copied using the procedure shown in Copying an angle. See that page for the proof. |
| 3 | Copied using the procedure shown in Copying an angle. See that page for the proof. | |
| 4 | m∠EAD = |
The three angles at A form a straight angle, so add to 180° |
| 5 | m∠ABC = |
Interior angles of a triangle add to 180° |
| 6 | Therefore m∠ABC = m∠EAD |
From (4) and (5), both equal to the same thing |
| 7 | m∠EAC = |
Exterior angle of a triangle is the sum of the opposite interior angles |
| 8 | Therefore m∠ACB = m∠CAD = given angle C |
From (3) and (7) by the transitive property of equality |
| 9 | The triangle has the given side and two angles | From (2), (8) and (1) |
- Q.E.D
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.
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.
Try it yourself
Click here for a printable worksheet containing two AAS 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
- Introduction to constructions
- Copy a line segment
- Sum of n line segments
- Difference of two line segments
- Perpendicular bisector of a line segment
- Perpendicular from a line at a point
- Perpendicular from a line through a point
- Perpendicular from endpoint of a ray
- Divide a segment into n equal parts
- Parallel line through a point (angle copy)
- Parallel line through a point (rhombus)
- Parallel line through a point (translation)
Angles
- Bisecting an angle
- Copy an angle
- Construct a 30° angle
- Construct a 45° angle
- Construct a 60° angle
- Construct a 90° angle (right angle)
- Sum of n angles
- Difference of two angles
- Supplementary angle
- Complementary angle
- Constructing 75° 105° 120° 135° 150° angles and more
Triangles
- Copy a triangle
- Isosceles triangle, given base and side
- Isosceles triangle, given base and altitude
- Isosceles triangle, given leg and apex angle
- Equilateral triangle
- 30-60-90 triangle, given the hypotenuse
- Triangle, given 3 sides (sss)
- Triangle, given one side and adjacent angles (asa)
- Triangle, given two angles and non-included side (aas)
- Triangle, given two sides and included angle (sas)
- Triangle medians
- Triangle midsegment
- Triangle altitude
- Triangle altitude (outside case)
Right triangles
- Right Triangle, given one leg and hypotenuse (HL)
- Right Triangle, given both legs (LL)
- Right Triangle, given hypotenuse and one angle (HA)
- Right Triangle, given one leg and one angle (LA)
Triangle Centers
Circles, Arcs and Ellipses
- Finding the center of a circle
- Circle given 3 points
- Tangent at a point on the circle
- Tangents through an external point
- Tangents to two circles (external)
- Tangents to two circles (internal)
- Incircle of a triangle
- Focus points of a given ellipse
- Circumcircle of a triangle
Polygons
- Square given one side
- Square inscribed in a circle
- Hexagon given one side
- Hexagon inscribed in a given circle
- Pentagon inscribed in a given circle
Non-Euclidean constructions
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