Squares
Each vertex looks the same - has the exact same composition of shapes around it:
four squares.
The interior angle of a square is 90 degrees.
90 + 90 + 90 + 90
= 360.
Therefore, squares tessellate!
Pentagons
Oh no! We have a gap between the pentagons!
There are three pentagons at the vertex.
The interior angle of a pentagon is 108 degrees.
108 + 108 + 108
= 324.
Not 360!
Therefore, pentagons do not tessellate.
Triangles
Each vertex looks the same - has the exact same composition of shapes around it:
six triangles.
The interior angle of an equilateral triangle is 60 degrees.
60 + 60 + 60 + 60 + 60 + 60
= 360.
Therefore, triangles tessellate!