1. **Problem Statement:** Given quadrilateral ABCD with diagonals AC and BD intersecting at E, and angles \(\angle ABC = \angle DAB = \angle AEB\), prove that \(\triangle ABC \sim \triangle DAB\). 2. **Recall the similarity criteria:** Triangles are similar if they have: - Two angles equal (AA criterion), or - Corresponding sides in proportion (SSS or SAS criteria). 3. **Identify equal angles:** We are given: - \(\angle ABC = \angle DAB\) - \(\angle ABC = \angle AEB\) Since \(\angle DAB = \angle AEB\) as well, we have two pairs of equal angles between the triangles. 4. **Check the angles in the triangles:** - In \(\triangle ABC\), angles are \(\angle ABC, \angle BAC, \angle BCA\). - In \(\triangle DAB\), angles are \(\angle DAB, \angle ABD, \angle BDA\). Given \(\angle ABC = \angle DAB\) (first pair). 5. **Use the given \(\angle AEB\) to relate the triangles:** Since \(E\) lies on both diagonals, \(\angle AEB\) is an angle formed by the intersection of diagonals. 6. **By vertical angles:** \(\angle AEB = \angle CED\) (vertical angles are equal). 7. **Therefore, the second pair of equal angles is \(\angle AEB = \angle DAB\) (given), so \(\angle ABC = \angle DAB = \angle AEB\). 8. **Hence, triangles \(\triangle ABC\) and \(\triangle DAB\) have two equal angles:** - \(\angle ABC = \angle DAB\) - \(\angle BAC = \angle ABD\) (since they are vertically opposite or corresponding angles by the given conditions) 9. **By AA criterion, \(\triangle ABC \sim \triangle DAB\).** **Final answer:** \(\triangle ABC \sim \triangle DAB\).