Constructing triangles with a compass – congruence criteria

What it shows

A triangle is fixed by three sides (SSS), two sides and the included angle (SAS), or one side and the two angles at its ends (ASA). In the ruler-and-compass construction for SSS the third vertex lies on two arcs centred at B and C, which meet in exactly two points that are mirror images across BC, so every triangle built from the same data is congruent. The construction fails when the triangle inequality is broken, because the arcs do not meet, or when angles B and C add up to 180° or more, because the rays never meet.

How to use

Choose a Construction case and set the sides and angles it uses. Click Next step to draw one stage at a time, asking students to predict each stage before it appears. After the last step, drag the green test point A and try to find any other position that fits the data. Then enter sides of 6, 2 and 3 cm, or angles B and C that total more than 180°, and click Redraw to see why no triangle can be built.

Parameters you can change

• Side BC = a 1–10 cm
• Side CA = b 1–10 cm
• Side AB = c 1–10 cm
• Construction case SSS – three sides a, b, c, SAS – BC = a, BA = c, angle B, ASA – BC = a, angle B, angle C
• Angle B (used for SAS, ASA) 10–150 °
• Angle C (used for ASA) 10–150 °

Questions to explore

1. Why are the two intersection points of the arcs mirror images of each other across BC?
2. Which side lengths make the two arcs fail to meet, and which inequality explains this?
3. What condition must angle B + angle C satisfy for the ASA construction to produce a triangle?