Can we construct a given angle again using only compass and ruler? 
The figure above shows an angle $\angle AOB$. Is it possible to construct angle $\angle COA = \angle AOB$ using just a compass and a ruler of infinite length ? ( initially we are given just $\angle AOB$ and have to construct $\angle COA$ )
 A: There is a nicely worded theorem from "A Readable Introduction to Real Mathematics" By Rosenthal$^3$ (there's three of them) which states: 
Any given angle can be copied using only a straightedge and compass 
proof Let an angle $ABC$ be given. We construct an angle equal to $\angle ABC$ with vertex $G$ on any other line. To do this, draw any arc of any circle (of radius, say, $r$) centered at $B$ that intersects both $BA$ and $BC$. Label the points of intersection $D$ and $E$. Draw the circles of radius $r$ centered at $G$. Use $H$ to label the point where that circle intersects the line containing $G$. Then adjust the compass to be able to make circles of radius $DE$. Put the point of the compass at $H$ and draw a portion of the circle that intersects the circle centered at $G$; call that point of the intersection $I$. Draw line segments connecting $D$ to $E$ and $I$ to $H$. 
Then $IH=DE$, since $IH$ is the radius of a circle with radius $DE$. Also draw line segment $GI$. The lengths of $BD, BE, GI$ and $GH$ are all equal to $r$. It follows by side-side-side that triangle $BDE$ is congruent to triangle $GIH$. Thus, $\angle IGH$ is a copy of $\angle ABC$.
