Constructions
63. The ASA Triangle Builder · Two angles and a side lock the shape in place
The rays always meet at C, forming a valid triangle with the given angles and side.
ASA construction (Angle-Side-Angle): given one side and the two angles at its endpoints, the triangle is uniquely determined. Draw the two angle rays at A and B; they always meet at C, completing △ABC.
What this lesson covers
Try to break it
What happens if angle A + angle B approaches 180°? The rays run parallel and never meet!
How you build it
Construct a triangle given two angles and a side.
- Mark point A — one end of the base.
- Mark point B — the given base length away from A.
- Draw the base AB.
- At A, set the angle dial to the first given angle and draw a ray from A measured from AB.
- At B, set the angle dial to the second given angle and draw a ray from B measured from BA.
- Mark point C where the two rays cross — the rays already form sides AC and BC, so triangle ABC is complete.
The proof, step by step
Prove that the construction produces the triangle fixed by the two angles and the included side.
- Draw the given side AB with the exact specified length.
- At vertex A, construct a ray that makes the given angle ∠A with side AB.
- At vertex B, construct a ray that makes the given angle ∠B with side AB.
- The point where these two rays intersect is vertex C. Join A, B, and C to form the unique triangle satisfying ASA conditions.
Worked example
Construct ΔPQR with PQ = 6 cm, ∠P = 50°, ∠Q = 70°. What is the measure of ∠R?
The sum of angles in any triangle is 180°. So, ∠R = 180° - (∠P + ∠Q) = 180° - (50° + 70°) = 180° - 120° = 60°.
- 50°
- 60° — correct
- 70°
- 80°