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Let the angle bisector of BAC intersect segment BC at point D.
Since ray AD is the angle bisector, angle BAD = angle CAD.
The segment AD = AD = itself.
Also, AB = AC since the triangle is isosceles.
Thus, triangle BAD is congruent to CAD by SAS Test.
Therefore triangle BAD = triangle CAD, and corresponding sides and angles are equal.
DB = DC,
angle ABD = angle ACD
angle ADB = angle ADC. From congruence of triangles, angle ADB = angle ADC. But by addition of angles, angle ADB + angle ADC = straight angle = 180 degrees. Thus 2 angle ADB = straight angle and angle AMB = 90 degrees = right angle.