We have seen that triangles do not exist for all sets of sidelengths. Is there a combination of measurements in the case of two sides and the included angle where a triangle is not possible? Justify your answer using what you observe during construction.
No, there is no combination of two sides and the included angle for which a triangle is not possible.
Justification: When two sides and the included angle are given, we can always construct the triangle.
Construction steps:
- Draw the base
- Construct the angle at one endpoint
- Mark the third vertex on the other arm of the angle
Why it works: Since the angle is included between the two sides, the two arms of the angle are fixed, and the third vertex can always be marked at the required distance.
Unlike the case of three sides where the arcs may not intersect, here the construction always succeeds.
Explanation
The chapter contrasts this with the case of three sides where the triangle inequality must be satisfied. When two sides and the included angle are given (SAS - Side-Angle-Side), the construction always works because: we draw one side as the base, construct the given angle at one endpoint, and the third vertex is uniquely determined by the intersection of the angle's other arm and the distance of the second side. There is no condition that can prevent this construction.
Solution Steps
Draw the base (one given side)
Construct the given angle at one endpoint
Mark the third vertex on the other arm of the angle at the required distance