copying an angle
Suppose someone has drawn an angle — two rays meeting at a vertex — and you want to reproduce exactly that opening somewhere else, at a new vertex pointing a new way. A protractor would let you read the degrees and redraw, but constructions forbid the protractor. The challenge is to copy an angle's size using only straightedge and compass, never reading a number.
The trick converts the angle into a length you can copy. Call the given angle one with vertex A and rays through P and Q. Draw an arc centred at A that crosses both rays, at points P and Q. Now draw a ray from the new vertex A', and an arc of the SAME radius centred at A' crossing it at P'. Set the compass to the straight-line distance |PQ| (the chord), place it at P', and swing an arc cutting the first arc at Q'. The ray A'Q' makes an angle equal to the original. In effect you copied the chord |PQ|, and equal chords on equal circles subtend equal angles.
Copying an angle is the partner of copying a segment, and together they let you reproduce any triangle (side-angle-side, angle-side-angle, and so on), draw a parallel by copying the angle a transversal makes, and build up complicated figures angle by angle. It quietly underlies the congruence constructions you meet right after learning SAS and ASA.
Angle at A with rays to P, Q: an arc gives points P, Q on the rays. At new vertex A' draw the same-radius arc hitting P', then step off the chord |PQ| from P' to land Q'. Ray A'Q' equals the original angle.
An angle is copied by copying the chord between its two arc-points.
The construction copies an angle's measure but says nothing about trisecting it: copying any angle is easy, yet dividing a general angle into three equal parts with straightedge and compass is impossible — a contrast worth keeping straight.