In this problem, we are given the following angles in the figure:

• $\angle FBC = 35^\circ$
• $\angle EDC = 34^\circ$
• $\angle FAE = 42^\circ$

We are tasked with finding the measure of $\angle FEA$.

Solution:

To find $\angle FEA$, we need to examine triangle $AFE$ closely.

• In triangle $AFE$, we know two angles: $\angle FAE = 42^\circ$ and $\angle FBC = 35^\circ$.
• Using the angle sum property of a triangle, which states that the sum of all angles in a triangle is $180^\circ$, we can find $\angle FEA$.

The sum of angles in triangle $AFE$ is:

$\angle FAE + \angle FEA + \angle AFE = 180^\circ$

Substitute the known angles:

$42^\circ + \angle FEA + 35^\circ = 180^\circ$

Simplifying this:

$77^\circ + \angle FEA = 180^\circ$

$\angle FEA = 180^\circ - 77^\circ = 103^\circ$

Answer:

The measure of $\angle FEA = 103^\circ$.

Do you need further details or have any other questions?

Here are some related questions to explore:

1. How do you verify the sum of angles in a triangle?
2. Can this problem be solved using alternate methods like exterior angles?
3. What are the properties of angles in a quadrilateral formed by such triangles?
4. How can this type of problem be applied to coordinate geometry?
5. How do angle bisectors affect the measurements in triangles like this?

Tip: Always double-check the sum of angles in a triangle when working through geometry problems to ensure accuracy.