1. **Problem statement:** We need to find the size of angle $m$ in a quadrilateral where two pairs of opposite sides are equal, and two angles are given: $142^\circ$ and $156^\circ$.\n\n2. Since the quadrilateral has two pairs of equal opposite sides, it is a parallelogram. Opposite angles of a parallelogram are equal, and adjacent angles sum to $180^\circ$.\n\n3. Given that one angle is $142^\circ$, the angle adjacent to it, which is $m$, satisfies:\n$$m + 142^\circ = 180^\circ$$\n\n4. Solve for $m$:\n$$m = 180^\circ - 142^\circ = 38^\circ$$\n\n5. Check opposite angle to $142^\circ$ is also $142^\circ$, and adjacent to $156^\circ$ is $m = 38^\circ$, confirming the parallelogram angle properties.