How many triangles can be formed for a triangle 𝐴𝐵𝐶 with 𝑏 equal to five centimetres, 𝑎 equal to two centimeters, and the measure of angle 𝐴 equal to 155 degrees? Is it (A) zero triangles, (B) one triangle, (C) two triangles, (D) three triangles, or (E) an infinite number of triangles?
Let’s begin by trying to sketch triangle 𝐴𝐵𝐶. We are told that the measure of angle 𝐴 is 155 degrees. This is an obtuse angle, since it lies between 90 and 180 degrees. We are told that side length 𝑎 is equal to two centimeters, and this must be opposite the 155-degree angle. We are also told that one of the other side lengths 𝑏 is equal to five centimeters. Looking at our triangle, this doesn’t appear to be possible as the five-centimeter side looks significantly shorter than the two-centimeter side. This is backed up by the fact that the longest side of any triangle will be opposite the largest angle.
Since angles in a triangle sum to 180 degrees, angle 𝐵 must be less than 25 degrees and is therefore smaller than angle 𝐴. As a result, side length 𝑏 would need to be smaller than side length 𝑎, and this is not the case. All of our working so far suggests that there are no triangles that can be formed from the information given. This can be more formally stated as follows. If angle 𝐴 is obtuse and side length 𝑎 is less than or equal to side length 𝑏, then no triangles exist. We can therefore conclude that the correct answer is option (A), zero triangles.