In triangles ABC and DEF, AC = DF, BC = EF, angle C = angle F. The bisectors of angles BAC and ABC intersect at point O

| February 4, 2021 | education

In triangles ABC and DEF, AC = DF, BC = EF, angle C = angle F. The bisectors of angles BAC and ABC intersect at point O, and the bisectors of angles DEF and EDF – at point M. Prove that triangle AOB = triangle DME.

First way.

In triangles ABC and DE, by condition, two sides and the angle between them are equal, then these triangles are equal by the first sign of equality of triangles.

In equal triangles, bisectors drawn from equal angles of triangles are equal, and at the point of their intersection they are divided into equal segments.

Then AO = DM, BO = EM.

Then the triangles AOB and DME are equal on three sides, as required.

Second way.

As we have proved that triangles ABC and DEF are equal, then the angle DEF = ABC, angle BAC = EDF.

Since the bisectors divide the angles in half, the angle ABO = DEM, the angle BAO = EDM.

Then, in triangles AOB and DME, side AB = EM and two adjacent angles are equal, then the triangles are equal in side and adjacent angles, which was required to prove.



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