In triangle \( ABC \), let \( \angle A = 58^\circ \) and \( \angle B = 96^\circ \). We need to find angle \( AMB \) where \( M \) is the intersection point of the angle bisectors of angles \( A \) and \( B \).
First, we can find angle \( C \) using the fact that the sum of angles in a triangle is \( 180^\circ \):
\[ \angle C = 180^\circ - \angle A - \angle B = 180^\circ - 58^\circ - 96^\circ = 26^\circ. \]
Next, we determine the angles at point \( M \). Since \( M \) is the intersection of the angle bisectors of angles \( A \) and \( B \), we can find angles \( AMB \) using the following properties of angle bisectors:
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The angle bisector of \( \angle A \) divides it into two equal angles: \[ \angle AMB = 90^\circ + \frac{\angle C}{2}. \]
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The angle bisector of \( \angle B \) also divides it into two equal angles, contributing another equivalent relationship that gives:
\[ \angle AMB = 90^\circ + \frac{\angle A}{2}. \] \[ \angle AMB = 90^\circ + \frac{\angle B}{2}. \]
However, we can use the first property to compute \( \angle AMB \):
To find \( \frac{\angle C}{2} \): \[ \frac{\angle C}{2} = \frac{26^\circ}{2} = 13^\circ. \]
Now using this in the formula for \( \angle AMB \): \[ \angle AMB = 90^\circ + 13^\circ = 103^\circ. \]
Thus, the measure of angle \( AMB \) is
\[ \boxed{103^\circ}. \]
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