bisecting an angle
Given an angle, you want the ray that splits it into two equal halves — without a protractor to read the degrees and halve them. This is the angular cousin of bisecting a segment, and like that construction it falls out of a couple of compass arcs and one straightedge line.
Let the angle have vertex A. Place the compass point at A and draw an arc crossing both sides of the angle, at points P and Q, so |AP| = |AQ|. Now centre the compass at P and draw an arc inside the angle; keep the same width, centre it at Q, and draw another arc crossing the first at a point R. Draw ray AR. It bisects the angle exactly. The reason: the figure APRQ has |AP| = |AQ| and |PR| = |QR| by construction, so triangles APR and AQR are congruent (SSS), forcing the two angles at A to be equal.
Bisecting an angle is everywhere downstream. The three angle bisectors of a triangle meet at the incentre, the centre of the inscribed circle. Repeated bisection lets you build 45-degree, 22.5-degree and 30-degree angles, and it is one of the standard moves for inscribing certain regular polygons. Note the sharp contrast with trisection: halving any angle is routine, but dividing a general angle into three is provably impossible with these tools.
Angle at A: an arc marks P and Q on its two sides. Equal arcs from P and from Q meet at R. Ray AR splits the angle into two equal parts, because triangles APR and AQR are congruent by SSS.
Two equal arcs and one ray cut any angle exactly in half — a congruent-triangle argument guarantees it.
Bisection is unlimited (you can halve again and again), but trisection is not: that any angle can be halved with straightedge and compass, yet a general angle cannot be trisected, is one of the great surprises of the subject.