To find the measure of angle $\angle AOB$, we can start by analyzing the geometry of the given problem.

Problem Analysis:

1. Square Properties:

   • The two circles are inscribed within the square and are tangent to each other.
   • The circles are also tangent to the sides of the square.

2. Points of Tangency:

   • Points $A$ and $B$ are points of tangency where the circles touch the sides of the square.
   • Point $O$ is the point where the two circles are tangent to each other.

Steps to Find the Angle:

1. Side Length and Radius:

   • Let $s$ be the side length of the square.
   • Let $r$ be the radius of each circle.
   • Since the circles are tangent to each other and to the sides of the square, the distance between the centers of the circles is $2r$.
   • Each circle's center is $r$ units away from the sides of the square.

2. Triangle and Diagonals:

   • The diagonal of the square is $s\sqrt{2}$.
   • In a square with side length $s$, the diagonal splits the square into two 45-45-90 triangles.

3. Angle Calculation:

   • Since the circles are tangent and centered at the midpoints of the sides they touch, the triangle $\triangle AOB$ forms a right-angled triangle at $O$.
   • The diagonals of the square meet at the center, forming four 90° angles.

However, we want the angle $\angle AOB$ determined by the lines connecting the centers of the circles to the tangency point $O$. Since the centers of the circles are also tangency points, we consider the symmetry and properties of the geometric setup.

By symmetry and considering the geometry of the circles within the square:

• The angles formed at the centers are straight lines.
• The tangency points and the centers create 45° angles due to the properties of tangency and the inscribed square.

Conclusion:

• The measure of angle $\angle AOB$ is 90 degrees because it represents a right angle formed by the tangents meeting the square's sides and the property of inscribed circles.

Would you like a detailed breakdown of any specific step? Here are some related questions for deeper understanding:

1. How do you calculate the diagonal of a square?
2. What are the properties of a 45-45-90 triangle?
3. How do the tangency points of the circles relate to the sides of the square?
4. How do you determine the distance between the centers of two tangent circles?
5. How does symmetry help in solving geometric problems involving squares and circles?
6. What are the key properties of a tangent line to a circle?
7. How can the Pythagorean theorem be used in this context?
8. How does the radius of the circles relate to the side length of the square?

Tip: When working with geometric figures, always look for symmetries and properties of standard shapes like squares and circles. These can simplify your calculations significantly.