1. **Stating the problem:** We have two similar isosceles triangles ABC and DAB with AB = AC and AD = BD. We know the ratio of sides BC to CD is 4:21. We need to find the ratio AB : AD. 2. **Understanding similarity and isosceles properties:** Since ABC and DAB are similar triangles, corresponding sides are proportional. Also, AB = AC in triangle ABC and AD = BD in triangle DAB. 3. **Assign variables:** Let AB = AC = x and AD = BD = y. 4. **Using the ratio BC : CD = 4 : 21:** Since BC and CD are sides opposite to vertices A and B respectively, and triangles are similar, the ratio of BC to CD corresponds to the ratio of sides in the triangles. 5. **Express BC and CD in terms of x and y:** In triangle ABC, BC is the base opposite vertex A. In triangle DAB, CD is the base opposite vertex B. Since triangles are similar, the ratio BC/CD = AB/AD = x/y. 6. **Set up the proportion:** $$\frac{BC}{CD} = \frac{x}{y} = \frac{4}{21}$$ 7. **Solve for the ratio AB : AD:** $$AB : AD = x : y = 4 : 21$$ **Final answer:** The ratio $AB : AD$ is $4 : 21$.