Constructions
62. SAS Triangle Builder · Two sides and the included angle
The triangle always satisfies the given side lengths and included angle.
SAS construction (Side-Angle-Side): given two sides and the included angle between them, the triangle is uniquely determined. Draw the included angle, mark the two sides on its arms, connect — one and only one triangle results.
What this lesson covers
Try to break it
Try dragging point A. Watch how side AB changes, but the angle at B stays locked at 60°. Can you make AB longer than BC – or so short the triangle nearly collapses?
How you build it
Construct a triangle given two sides and an angle.
- Mark point B — the vertex of the included angle.
- Mark point C — one given side length away from B.
- Draw side BC.
- At B, set the angle dial to the given angle and draw a ray from B measured from BC. This ray is the arm that will carry side BA.
- With centre B, open the compass to the second given side and draw an arc cutting the ray.
- Mark point A where the arc crosses the ray — side BA is now complete.
- Draw side AC to complete triangle ABC.
The proof, step by step
Prove that the construction produces the triangle fixed by the two sides and the included angle.
- Draw the base BC with the given length.
- Construct the given included angle at vertex B.
- Mark point A on the ray such that BA equals the given length.
- Join A and C to complete the triangle.
Worked example
To construct a triangle ABC given AB = 5 cm, BC = 7 cm and ∠B = 50°, which side should be drawn first?
In SAS construction, we conventionally draw the side connecting the two given sides (BC) first, then construct the included angle at B, and finally mark the second side (AB) along the angle's ray.
- BC = 7 cm — correct
- AB = 5 cm
- AC first
- Any side works equally well