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The problem requires finding the measure of angle \( \angle FHE \) in degrees, given a circle with points \( F \), \( H \), \( E \), \( A \), and \( L \) on its circumference.

### Given:
- \( \angle LFA = 65.8^\circ \)
- \( \angle LAE = 23^\circ \)

### Solution:
The problem is based on properties of cyclic quadrilaterals and circle theorems.

1. **Inscribed Angle Theorem**: The angle subtended by an arc at the center of a circle is twice the angle subtended at any point on the circumference by the same arc. Since all points lie on the circle, the angles can be related by the arcs they subtend.

2. **Exterior Angle of a Cyclic Quadrilateral**: The exterior angle of a cyclic quadrilateral is equal to the interior opposite angle.

Given \( \angle FHE \) as the exterior angle of quadrilateral \( AFHE \):

\[
\angle FHE = \angle LFA + \angle LAE
\]

\[
\angle FHE = 65.8^\circ + 23^\circ = 88.8^\circ
\]

Thus, the measure of \( \angle FHE \) is **88.8 degrees**.

Would you like further details or have any questions?

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Here are some related questions you might consider:
1. How do you calculate angles using the Inscribed Angle Theorem?
2. What properties do cyclic quadrilaterals have?
3. How do you identify and calculate exterior angles in cyclic quadrilaterals?
4. What are the applications of the Inscribed Angle Theorem in different geometric problems?
5. How can you verify your solution using other geometric principles?

**Tip:** Always check if the points given in a problem lie on a circle, as this can often simplify the problem using circle theorems.

Find the measure of angle FHE in degrees, given a circle with points F, H, E, A, and L on its circumference, where angle LFA = 65.8 degrees and angle LAE = 23 degrees.

Learn how to calculate the measure of angle FHE in a circle with points on its circumference using circle theorems. Understand the Inscribed Angle Theorem and the Exterior Angle Theorem for Cyclic Quadrilaterals to find the solution. Suitable for students in Grades 9-12.

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