The diagonals of rectangle ABCD intersect at a point O. If ∠COD is 78°, then find ∠OAB.

1. Break Down the Problem

To find angle ∠OAB in rectangle ABCD, we can utilize the properties of the rectangle and the relationships between angles formed by intersecting lines.

2. Relevant Concepts

In any rectangle:

• The diagonals bisect each other, which means that: $\angle AOB = \angle COD = 78^\circ$
• The diagonals of a rectangle are equal and divide the rectangle into two congruent triangles. Hence, the angles around point O sum to $360^\circ$.

3. Analysis and Detail

1. Since $\angle AOB$ and $\angle COD$ are vertical angles, they are equal: $\angle AOB = 78^\circ$
2. Considering quadrilateral AOBD (formed by the diagonals and sides of rectangle ABCD):
   • The sum of angles around point O equals $360^\circ$: $\angle AOB + \angle BOC + \angle COD + \angle DOA = 360^\circ$
   • Since diagonal AC = diagonal BD, triangles AOB and COD are congruent: $\angle AOB + \angle OAB + \angle OAD = 180^\circ \quad \text{(as they form a straight line)}$ Thus, $\angle OAB + 78^\circ + 78^\circ = 180^\circ$

4. Verify and Summarize

Solving for ∠OAB: $\angle OAB + 156^\circ = 180^\circ$ $\angle OAB = 180^\circ - 156^\circ = 24^\circ$

Final Answer

Therefore, the angle ∠OAB is: $\boxed{24^\circ}$