Geometry Problem Solver How to bisect an angle
To bisect an angle is to cut it into two equal angles. With a compass and straightedge: draw an arc from the vertex crossing both arms, then from each crossing point draw an arc so the two meet, and finally rule a line from the vertex through that meeting point. No measuring is needed.
The construction, step by step
Step 1. Put the compass point on the vertex and draw an arc wide enough to cross both arms of the angle. Mark the two crossing points.
Step 2. Move the compass point to one of those crossings and draw an arc into the middle of the angle. Do not change the compass width from here on.
Step 3. Move the point to the other crossing and draw a second arc that cuts the first. Mark where they intersect.
Step 4. Rule a straight line from the vertex through that intersection. That line is the bisector, and the two angles either side of it are equal.
Why it works
The first arc puts the two crossing points at an equal distance from the vertex. The two matching arcs put the meeting point at an equal distance from each crossing. So the bisector and the two arms form two triangles with three pairs of equal sides.
Triangles with three equal sides are congruent, by the side-side-side rule, and congruent triangles have equal angles at the vertex. That is the proof, and it is why the method needs no protractor: equality is built by construction rather than measured.
What usually goes wrong
Changing the compass width between steps 2 and 3. The two arcs then meet somewhere off the true bisector and the halves come out unequal.
Drawing the first arc too small, so it crosses one arm but misses the other, or crosses so close to the vertex that the marks are hard to hit accurately.
Ruling from the intersection of the arcs without going through the vertex. The bisector of an angle has to start at the vertex.
Questions
Can I just measure the angle and halve it?
For a drawing, yes, and for a 90° angle it is faster. Geometry courses ask for the compass construction because it is exact, while a protractor reading is only as good as your eyesight, and because the congruent-triangle proof behind it is the point of the exercise.
How do I bisect a reflex angle?
Bisect the smaller angle on the other side, then extend that bisector straight through the vertex and out the far side. The extension is the bisector of the reflex angle.
What is the difference between bisecting an angle and bisecting a line?
An angle bisector splits an angle into two equal angles and passes through the vertex. A perpendicular bisector splits a segment into two equal lengths and crosses it at a right angle. The constructions look similar because both rely on equal compass distances.
More geometry reference
Stuck on the actual problem
Photograph it and get the worked solution, step by step, with the theorem named at each one.
