Constructions and loci

1. (a) Step 1: arc centred on $B$ crossing both arms, at $X$ and $Y$.
2. Step 2: equal arcs from $X$ and $Y$ that cross.
3. Step 3: join $B$ to the crossing point.
4. (b) Equal radii make the crossing point equidistant from $X$ and $Y$, and $BX = BY$, so the line from $B$ through it is a line of symmetry of the angle.
5. (c) With the compass point on $B$, draw an arc that crosses $BA$ at $X$ and $BC$ at $Y$.
6. Keep the compasses at one radius. Draw an arc from $X$ and an arc from $Y$ so that they cross at $Z$, inside the angle.
7. Draw the line from $B$ through $Z$. Leave every arc visible. Check with a protractor: the two halves should each be about $35^\circ$.

Answer (a) With the compass point on $B$, draw an arc crossing both arms of the angle.; (b) The crossing point must be the same distance from both points, so it lies on the line of symmetry of the angle.; (c) An arc centred on $B$ crossing both arms at $X$ and $Y$; equal arcs from $X$ and $Y$ crossing at $Z$; the line $BZ$, with every arc left visible.