Given that triangles \( \triangle ABC \) and \( \triangle SRT \) are similar (\( \triangle ABC \sim \triangle SRT \)), the corresponding angles are equal.
From the information given:
- In triangle \( ABC \):
- \( m\angle A = a = 90^\circ \)
- \( m\angle C = c = 34^\circ \)
To find \( m\angle B \), we can use the fact that the sum of the angles in a triangle is \( 180^\circ \): \[ m\angle A + m\angle B + m\angle C = 180^\circ \]
Substituting the known values: \[ 90^\circ + m\angle B + 34^\circ = 180^\circ \] \[ m\angle B + 124^\circ = 180^\circ \] \[ m\angle B = 180^\circ - 124^\circ \] \[ m\angle B = 56^\circ \]
Since \( \triangle ABC \sim \triangle SRT \), the corresponding angle \( m\angle B \) in triangle \( ABC \) corresponds to angle \( m\angle R \) in triangle \( SRT \). This is consistent with the provided information:
- \( m\angle R = 56^\circ \)
Thus: \[ \boxed{56} \]