To solve for $\angle BDC$ and $\angle ABE$, let's analyze the information given in the diagram and the properties of the circle and the angles.

Step-by-Step Solution

1. Understanding Given Information:

   • $ABCD$ is a circle.
   • $E$ is a point on line $AC$ such that $BC = CE$.
   • $AB = AD$.
   • $\angle ADB = 58^\circ$ and $\angle CBD = 25^\circ$.

2. Finding $\angle BDC$:

   • Since $ABCD$ is a circle, and points $B$, $D$, and $C$ are on the circumference, we can use the property of a cyclic quadrilateral.
   • In a cyclic quadrilateral, the opposite angles add up to $180^\circ$.
   • Thus, $\angle ADB + \angle BDC = 180^\circ$.
   • Given $\angle ADB = 58^\circ$, we have: $\angle BDC = 180^\circ - 58^\circ = 122^\circ$

3. Finding $\angle ABE$:

   • Since $BC = CE$, triangle $BCE$ is isosceles.
   • Therefore, $\angle CBE = \angle CEB$.
   • We are given $\angle CBD = 25^\circ$, which means $\angle CBE = 25^\circ$ (as $\angle CBE$ and $\angle CBD$ are the same in this isosceles setup).
   • $\angle ABE$ is the external angle for $\triangle BCE$ at point $B$, so: $\angle ABE = \angle CBD + \angle CBE = 25^\circ + 25^\circ = 50^\circ$

Final Answers

• $\angle BDC = 122^\circ$
• $\angle ABE = 50^\circ$

Would you like a detailed explanation of the concepts used, or do you have any questions?

Here are some related questions that expand on this topic:

1. How does the property of opposite angles in a cyclic quadrilateral help in solving circle geometry problems?
2. What are the properties of an isosceles triangle that are useful in circle geometry?
3. How would you determine angles in a cyclic quadrilateral if no values were given initially?
4. Can external angles of polygons within circles help determine unknown internal angles?
5. What other properties of cyclic quadrilaterals can be applied in solving complex geometric problems?

Tip: In a circle, knowing a few properties of cyclic quadrilaterals can greatly simplify finding unknown angles. Look for opposite angles and isosceles triangles to leverage symmetry in your solutions.