To resolve the discrepancy, let's re-examine the problem with precise calculations:\newline
\newline
**Given**:\newline
- \( \triangle ABC \) is isosceles with \( AC = BC \) and \( \angle C = 20^\circ \).\newline
- \( \angle BAD = 25^\circ \), \( \angle ABE = 65^\circ \).\newline
- Points \( D \) on \( BC \) and \( E \) on \( AC \).\newline
\newline
**Steps**:\newline
1. **Base Angles**:\newline
   - \( \angle A = \angle B = 80^\circ \) (since \( \angle C = 20^\circ \)).\newline
\newline
2. **Coordinate Geometry**:\newline
   - Place \( C \) at the origin \((0,0)\), \( A \) at \((\cos10^\circ, \sin10^\circ)\), and \( B \) at \((\cos10^\circ, -\sin10^\circ)\).\newline
   - Determine coordinates for \( D \) and \( E \) using trigonometric relationships.\newline
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3. **Key Calculations**:\newline
   - For \( D \) on \( BC \), solving for \( t \) gives \( t \approx 0.846 \), leading to \( D \approx (0.833, -0.147) \).\newline
   - For \( E \) on \( AC \), solving for \( s \) gives \( s \approx 0.451 \), leading to \( E \approx (0.444, 0.0783) \).\newline
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4. **Vectors and Angle**:\newline
   - Vectors \( \vec{EB} \) and \( \vec{ED} \) yield:\newline
     \[
     \cos\theta \approx 0.997 \implies \theta \approx \arccos(0.997) \approx 4.4^\circ
     \]
\newline
**Conclusion**:\newline
The measure of \( \angle BED \) is approximately \( \boxed{5^\circ} \). The earlier assertion of \( 30^\circ \) was incorrect due to misapplied geometric assumptions. The coordinate method confirms the angle is small, aligning with your measurement.