To find the measure of the larger angle formed by the intersection of the angle bisectors of angles \( A \) and \( C \) in triangle \( ABC \), we start with the degrees of angles provided in the problem:
- \( \angle A = 58^\circ \)
- \( \angle C = 46^\circ \)
First, we can find the measure of angle \( B \) using the angle sum property of triangles, which states that the sum of the angles in a triangle is \( 180^\circ \):
\[ \angle B = 180^\circ - \angle A - \angle C \]
Substituting in the known values:
\[ \angle B = 180^\circ - 58^\circ - 46^\circ \]
Calculating the subtraction:
\[ \angle B = 180^\circ - 104^\circ = 76^\circ \]
Now, we have the measures of all three angles:
- \( \angle A = 58^\circ \)
- \( \angle B = 76^\circ \)
- \( \angle C = 46^\circ \)
Next, we need to determine the angles formed by the intersection of the angle bisectors of \( \angle A \) and \( \angle C \).
The angle bisector of angle \( A \) divides it into two angles of \( \frac{\angle A}{2} \):
\[ \text{Angle from bisector of } A = \frac{58^\circ}{2} = 29^\circ \]
Similarly, the angle bisector of angle \( C \) divides it into two angles of \( \frac{\angle C}{2} \):
\[ \text{Angle from bisector of } C = \frac{46^\circ}{2} = 23^\circ \]
When the angle bisectors intersect, they form two angles at the intersection point. The angle between the two angle bisectors, denoted as \( \angle D \), can be computed as follows:
\[ \angle D = \text{Angle from bisector of } A + \text{Angle from bisector of } C \]
Calculating this gives:
\[ \angle D = 29^\circ + 23^\circ = 52^\circ \]
The angle formed by the two angle bisectors with respect to the interior of the angle between them is \( 52^\circ \). However, the angles around point \( D \) sum to \( 360^\circ \), and the two other angles formed by the intersection line will be:
\[ 180^\circ - \angle D = 180^\circ - 52^\circ = 128^\circ \]
Thus, the measure of the larger angle formed by the intersection of these bisectors is:
\[ \boxed{128^\circ} \]