Proof:

Let's consider a figure with two intersecting lines, where angle b is smaller than angle a and angle c is smaller than angle d. We need to show that ad is smaller than bd.

Step 1: Angle Relationships
- Angle b is smaller than angle a: This implies that the measure of angle b is less than the measure of angle a.
- Angle c is smaller than angle d: This implies that the measure of angle c is less than the measure of angle d.

Step 2: Vertical Angles
- In the given figure, angle a and angle c are vertical angles, as they are formed by two intersecting lines. Vertical angles are congruent, meaning they have equal measures.

Step 3: Triangle ABC
- Consider triangle ABC, where angle a is an interior angle.
- According to the Triangle Angle Sum Theorem, the sum of the measures of the interior angles of a triangle is always 180 degrees.
- Therefore, angle b + angle a + angle c = 180 degrees.
- Since angle b is smaller than angle a and angle c is smaller than angle d, it follows that angle b + angle a + angle c < angle="" a="" +="" angle="" a="" +="" angle="" />
- Simplifying the inequality, we get angle b + angle a + angle c < 2angle="" a="" +="" angle="" />

Step 4: Exterior Angle
- Angle ad is an exterior angle of triangle ABC, formed by extending side BA.
- According to the Exterior Angle Theorem, the measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles.
- Therefore, angle ad = angle b + angle a.

Step 5: Conclusion
- From step 3, we have angle b + angle a + angle c < 2angle="" a="" +="" angle="" />
- Substituting angle ad for angle b + angle a, we get angle ad + angle c < 2angle="" a="" +="" angle="" />
- Since angle ad is smaller than angle ad + angle c, it follows that angle ad < 2angle="" a="" +="" angle="" />
- From the inequality, we can conclude that ad is smaller than bd, as desired.

Therefore, we have shown that in the given figure, where angle b is smaller than angle a and angle c is smaller than angle d, ad is smaller than bd.