Analyze Triangle Geometry Problem

We are given a triangle $\triangle ABC$. A line segment $BD$ is perpendicular to $AC$ at point D ($BD \perp AC$). We know the angle $\angle DBC = 71^\circ$. Another point E lies on the side BC, and we are given $\angle CAE = 17^\circ$. The goal is to find the measure of the angle $\angle AEB$.

Calculate Angle BCA

Consider the triangle $\triangle BDC$. Since $BD \perp AC$, the angle $\angle BDC$ is a right angle ($90^\circ$). The sum of angles in any triangle is $180^\circ$. In $\triangle BDC$, we have:

• $\angle BDC = 90^\circ$
• $\angle DBC = 71^\circ$

Using the angle sum property:

$ \angle BCA = 180^\circ - \angle BDC - \angle DBC $

$ \angle BCA = 180^\circ - 90^\circ - 71^\circ $

$ \angle BCA = 19^\circ $

Since point E lies on BC, $\angle ACE$ is the same angle, so $\angle ACE = 19^\circ$.

Determine Angle AEB using Triangle ABE

Let's denote the angle $\angle BAE$ as $\alpha$.

The angle $\angle BAC$ is the sum of $\angle BAE$ and $\angle CAE$:

$ \angle BAC = \angle BAE + \angle CAE $

$ \angle BAC = \alpha + 17^\circ $

Now, consider the larger triangle $\triangle ABC$. The sum of its angles must be $180^\circ$. We can find $\angle ABC$:

$ \angle ABC = 180^\circ - \angle BAC - \angle BCA $

Substitute the expressions for $\angle BAC$ and the value of $\angle BCA$:

$ \angle ABC = 180^\circ - (\alpha + 17^\circ) - 19^\circ $

$ \angle ABC = 180^\circ - \alpha - 17^\circ - 19^\circ $

$ \angle ABC = 180^\circ - \alpha - 36^\circ $

$ \angle ABC = 144^\circ - \alpha $

Since E is on BC, $\angle ABE$ is the same as $\angle ABC$. So, $\angle ABE = 144^\circ - \alpha$.

Finally, apply the angle sum property to triangle $\triangle ABE$:

$ \angle BAE + \angle ABE + \angle AEB = 180^\circ $

Substitute the known values and expressions:

$ \alpha + (144^\circ - \alpha) + \angle AEB = 180^\circ $

The $\alpha$ terms cancel out:

$ 144^\circ + \angle AEB = 180^\circ $

Solve for $\angle AEB$:

$ \angle AEB = 180^\circ - 144^\circ $

$ \angle AEB = 36^\circ $

Conclusion

The measure of angle $\angle AEB$ is $36^\circ$.