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##).
