Is it possible for a pair of nonadjacent angles to share vertex a and arm ab?

Yes: by definition, two adjacent angles have the same vertex, a side in common and don't overlap.

So, if two angles share the vertex and a side, but they overlap, they are not adjacent.

For example, consider the origin, the x and y axes, and the bisector of the first quadrant

Define angle between the x axis and the bisector, and angle between the x axis and the y axis. So, and have the same vertex (the origin) and one side in common (the x axis), but they do overlap, and are not adjacent.

Although your question is incomplete a general answer is provided :

No A pair of non-adjacent angles does not share the same vertex ( a )and a common side ( arm ab ) simultaneously

Non-adjacent angles are angles that do not share a common side and a common vertex simultaneously, but adjacent angles are angles that share a common vertex, common side and do not overlap each other. for angles to be non-adjacent one of the following must be missing :

Hence from your question a pair of nonadjacent angles will not share a vertex a and common side ( arm ab )

learn more :

let's add {f(x)=x+1}f(x)=x+1 and {g(x)=2x}g(x)=2x together to make a new function.

f(x)+g(x) =(x+1)+(2x)=x+1+2x=3x+1

let's call this new function hh. so we have:

{h(x)}={f(x)}+{g(x)}{=3x+1}h(x)=f(x)+g(x)=3x+1

we can also evaluate combined functions for particular inputs. let's evaluate function hh above for x=2x=2. below are two ways of doing this.

method 1: substitute x=2x=2 into the combined function hh.

h(x)

h(2)

​    

=3x+1

=3(2)+1

=7

​ since h(x)=f(x)+g(x)h(x)=f(x)+g(x), we can also find h(2)h(2) by finding f(2) +g(2)f(2)+g(2).

first, let's find f(2)f(2):

f(x)

f(2)

​    

=x+1

=2+1

=3

​  

now, let's find g(2)g(2):

g(x)

g(2)

​    

=2x

=2⋅2

=4

​  

so f(2)+g(2)=3+4=\greend7f(2)+g(2)=3+4=7.

notice that substituting x =2x=2 directly into function hh and finding f(2) + g(2)f(2)+g(2) gave us the same answer!