Is the notion of circle necessary for proving some problem of angle? As shown in the image for a plane geometric problem: 
Could we prove $\angle ACD=\angle ABD$ without using the notion of circle?
It could seem easy if we have the notion of circle. But if we have no the notion of circle?
 A: From $C$ draw $CE$ such that$$\angle  BCE=\angle CBA$$Hence$$BE=CE$$Then since$$\angle CAB+\angle CBA=\angle ACB=90^o$$complements$$\angle CAB=\angle ACE$$and$$CE=AE$$making$$BE=CE=AE$$Thus $E$ is the midpoint of $AB$, and by the same construction and argument, since $\triangle ABD$, like $\triangle ABC$, is any right triangle with hypotenuse $AB$, a line from $D$ making with $DB$ an angle equal to $\angle DBA$ passes through the midpoint of $AB$, making$$BE=DE=AE$$and hence$$\angle EAD=\angle EDA$$

Now consider $\angle ABD$ and $\angle ACD$:
Since $\angle ADB=90^o$ $$\angle ABD=90^o-\angle BAD$$
And$$\angle ACD=180^o-(\angle CAD+\angle CDA)=180^o-(\angle CAE+\angle CDE+2\angle EAD)$$that is, because of isosceles triangles $CEA$ and $CED$ $$\angle\ ACD=180^o-\angle ACD-2\angle EAD$$ Adding $\angle ACD$ to both sides$$2\angle ACD=180^o-2\angle EAD$$and dividing by $2$ $$\angle ACD=90^o-\angle EAD=90^o-\angle BAD=\angle ABD$$
This argument appears not to rest on any truths about the circle, although the auxiliary construction requires constructing angles equal to given $\angle ABC$ and $\angle ABD$, which in Euclid's treatment (Elements I, 23) does require drawing circles. Indeed,  except for the drawing and extension of straight lines, it seems for Euclid all construction requires drawing circles, and arguments based on those constructions must appeal at least to the defining property of the circle, that all its radii are equal. 
A: It all depends on how you define angle. If you could propose some alternate definition then that might be interesting. But, if you wish to use the standard definition of angle then you're stuck with circles.
Here is the standard definition of the radian measure of an angle. 
Given a plane $P$, a point $O$ in that plane, and a pair of points $X,Y \ne O$ in that same plane, the radian measure of the angle $XOY$ is defined as follows. Let $C$ be the circle of radius $1$ in $P$ centered at $O$. Let $x$ be the point where the ray $OX$ cuts through $C$. Let $y$ be the point where the ray $OY$ cuts through $C$. The points $x,y$ cut the circle $C$ into two circular arcs $\alpha$ and $\beta$. The angle $XOY$ is the minimum of the lengths of $\alpha$ and $\beta$.
As you can see, in this definition circles are at the heart of the very definition of angle. So if you had no notion of circle, then you would not have this definition of angle.
