Question

How do you bisect an obtuse angle?

Answer 1

Any angle, including obtuse, can be bisected by constructing congruent triangles with common side lying on an angle's bisector.
See details below.

Explanation:

Given angle `/_ABC` with vertex `B` and two sides `BA` and `BC`. It can be acute or obtuse, or right - makes no difference.

Choose any segment of some length `d` and mark point `M` on side `BA` on a distance `d` from vertex `B`.
Using the same segment of length `d`, mark point `N` on side `BC` on distance `d` from vertex `B`.
Red arc on a picture represents this process, its ends are `M` and `N`.

We can say now that `BM~=BN`.

Choose a radius sufficiently large (greater than half the distance between points `M` and `N`) and draw two circles with centers at points `M` and `N` of this radius. These two circles intersect in two points, `P` and `Q`. See two small arcs intersecting on a picture, their intersection is point `P`.

Chose any of these intersection points, say `P`, and connect it with vertex `B`. This is a bisector of an angle `/_ABC`.

Proof

Compare triangles `Delta BMP` and `Delta BNP`.
1. They share side `BP`
2. `BM~=BN` by construction, since we used the same length `d` to mark both points `M` and `N`
3. `MP~=NP` by construction, since we used the same radius of two intersecting circles with centers at points `M` and `N`.
Therefore, triangles `Delta BMP` and `Delta BNP` are congruent by three sides:
`Delta BMP ~=Delta BNP`

As a consequence of congruence of these triangles, corresponding angles have the same measure.
Angles `/_MBP` and `/_NBP` lie across congruent sides `MP` and `NP`.
Therefore, these angles are congruent:
`/_MBP ~= /_NBP`,
that is `BP` is a bisector of angle `/_MBP` (which is the same as angle `/_ABC`).