How do you bisect an obtuse angle?

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

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 ##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##).