The problem asks for the measure of angle AEB in a triangle ABC with specific conditions.

Geometric Analysis of Triangle ABC

We are given a triangle ABC where BD is perpendicular to AC at point D. This means $\angle\text{BDC} = 90^{\circ}$.

We are also given that $\angle\text{DBC} = 60^{\circ}$.

Calculating Angles in Triangle BDC

Consider the right-angled triangle BDC:

• The sum of angles in a triangle is $180^{\circ}$.
• In $\triangle\text{BDC}$, we have $\angle\text{BDC} + \angle\text{DBC} + \angle\text{BCD} = 180^{\circ}$.
• Substituting the known values: $90^{\circ} + 60^{\circ} + \angle\text{BCD} = 180^{\circ}$.
• Solving for $\angle\text{BCD}$: $150^{\circ} + \angle\text{BCD} = 180^{\circ}$, which gives $\angle\text{BCD} = 180^{\circ} - 150^{\circ} = 30^{\circ}$.
• Therefore, $\angle\text{BCA} = 30^{\circ}$.

Calculating Angles in Triangle ABC

Now, consider the larger triangle ABC:

• The sum of angles in $\triangle\text{ABC}$ is $180^{\circ}$: $\angle\text{ABC} + \angle\text{BCA} + \angle\text{CAB} = 180^{\circ}$.
• Substitute $\angle\text{BCA} = 30^{\circ}$: $\angle\text{ABC} + 30^{\circ} + \angle\text{CAB} = 180^{\circ}$.
• This simplifies to $\angle\text{ABC} + \angle\text{CAB} = 150^{\circ}$.

Finding Angle AEB

We are given a point E on BC such that $\angle\text{CAE} = 20^{\circ}$. We need to find $\angle\text{AEB}$.

Consider the triangle ABE:

• The sum of angles in $\triangle\text{ABE}$ is $180^{\circ}$: $\angle\text{AEB} + \angle\text{ABE} + \angle\text{BAE} = 180^{\circ}$.
• Note that $\angle\text{ABE}$ is the same as $\angle\text{ABC}$.
• The angle $\angle\text{BAE}$ can be expressed as $\angle\text{CAB} - \angle\text{CAE}$.
• So, $\angle\text{BAE} = \angle\text{CAB} - 20^{\circ}$.
• Substitute these into the angle sum for $\triangle\text{ABE}$: $\angle\text{AEB} + \angle\text{ABC} + (\angle\text{CAB} - 20^{\circ}) = 180^{\circ}$.
• Rearrange the terms: $\angle\text{AEB} + (\angle\text{ABC} + \angle\text{CAB}) - 20^{\circ} = 180^{\circ}$.
• We already found that $\angle\text{ABC} + \angle\text{CAB} = 150^{\circ}$.
• Substitute this value: $\angle\text{AEB} + 150^{\circ} - 20^{\circ} = 180^{\circ}$.
• Simplify: $\angle\text{AEB} + 130^{\circ} = 180^{\circ}$.
• Solve for $\angle\text{AEB}$: $\angle\text{AEB} = 180^{\circ} - 130^{\circ} = 50^{\circ}$.

Thus, the measure of angle AEB is $50^{\circ}$.