A scalene triangle ABC tiles the plane. What is a possible configuration of the angles at the vertices?
I: three types of vertices: 6 As, or 6 Bs, or 6 Cs
II: one type of vertex: 2 As, 2 Bs, and 2 Cs
A) I only
B) II only
C) both I and II
D) neither I nor II

Question

A scalene triangle ABC tiles the plane. What is a possible configuration of the angles at the vertices? 
I: three types of vertices: 6 As, or 6 Bs, or 6 Cs
II: one type of vertex: 2 As, 2 Bs, and 2 Cs
A) I only
B) II only
C) both I and II
D) neither I nor II

Quizplus · Accepted Answer

In a scalene triangle, each angle must be different from the other two angles. Therefore, in configuration I, if there are 6 As, each A must have a different angle measure, which is impossible because there are only two other angles to work with. The same applies for 6 Bs or 6 Cs.

On the other hand, in configuration II, it is possible to have a scalene triangle with 2 As, 2 Bs, and 2 Cs, where each angle is different from the other two angles. For example, the triangle could have angles of 50, 80, and 50 degrees for the As, 60, 80, and 40 degrees for the Bs, and 70, 70, and 40 degrees for the Cs. Therefore, the correct answer is B) II only.

A Scalene Triangle ABC Tiles the Plane