Question

Angle bisectors of angle A and angle B of a triangle ABC intersect each other at point M. Find angle AMB if angle A is 58 degrees and angle B is 96 degrees.

Answer 1

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:

1. The angle bisector of \( \angle A \) divides it into two equal angles:
\[
\angle AMB = 90^\circ + \frac{\angle C}{2}.
\]

2. 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

\[
\boxed{103^\circ}.
\]