#### Answer

$61^{\circ}$

#### Work Step by Step

With isosceles triangles, we need to remember that the base angles are congruent. If the vertex angle is $58^{\circ}$, then we can find the measurement of each of the base angles by using the triangle sum theorem, which states that the sum of the measures of the angles of a triangle equals $180^{\circ}$:
measure of the base angles = $180^{\circ} - (58^{\circ})$
Subtract to solve:
measure of the base angles = $122^{\circ}$
Since the base angles are congruent, each will be half of $122^{\circ}$. Let's find the measure of just one of the base angles:
measure of each base angle = $122^{\circ}/2$
Divide to solve:
measure of each base angle = $61^{\circ}$